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Theorem tfinnnlem1 4534

Description: Lemma for tfinnn 4535. Establish stratification. (Contributed by SF, 30-Jan-2015.)

**Assertion**
Ref	Expression
tfinnnlem1	⊢ {n ∣ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)} ∈ V

Distinct variable group: y,n,a,x

Proof of Theorem tfinnnlem1 Dummy variables t w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 ⊢ n ∈ V
2	1	eluni1 4174	. . . 4 ⊢ (n ∈ ⋃₁ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ {n} ∈ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c))
3		snex 4112	. . . . . . . . . 10 ⊢ {{y}} ∈ V
4		opkeq1 4060	. . . . . . . . . . 11 ⊢ (t = {{y}} → ⟪t, {n}⟫ = ⟪{{y}}, {n}⟫)
5	4	eleq1d 2419	. . . . . . . . . 10 ⊢ (t = {{y}} → (⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ⟪{{y}}, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
6	3, 5	ceqsexv 2895	. . . . . . . . 9 ⊢ (∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ ⟪{{y}}, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))))
7		elin 3220	. . . . . . . . 9 ⊢ (⟪{{y}}, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ (⟪{{y}}, {n}⟫ ∈ SI_k S_k ∧ ⟪{{y}}, {n}⟫ ∈ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))))
8		snex 4112	. . . . . . . . . . . 12 ⊢ {y} ∈ V
9	8, 1	opksnelsik 4266	. . . . . . . . . . 11 ⊢ (⟪{{y}}, {n}⟫ ∈ SI_k S_k ↔ ⟪{y}, n⟫ ∈ S_k )
10		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
11	10, 1	elssetk 4271	. . . . . . . . . . 11 ⊢ (⟪{y}, n⟫ ∈ S_k ↔ y ∈ n)
12	9, 11	bitri 240	. . . . . . . . . 10 ⊢ (⟪{{y}}, {n}⟫ ∈ SI_k S_k ↔ y ∈ n)
13		eldif 3222	. . . . . . . . . . 11 ⊢ (⟪{{y}}, {n}⟫ ∈ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ (⟪{{y}}, {n}⟫ ∈ SI_k (𝒫₁𝒫 Nn ×_k V) ∧ ¬ ⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))
14	8, 1	opksnelsik 4266	. . . . . . . . . . . . 13 ⊢ (⟪{{y}}, {n}⟫ ∈ SI_k (𝒫₁𝒫 Nn ×_k V) ↔ ⟪{y}, n⟫ ∈ (𝒫₁𝒫 Nn ×_k V))
15	8, 1	opkelxpk 4249	. . . . . . . . . . . . . . 15 ⊢ (⟪{y}, n⟫ ∈ (𝒫₁𝒫 Nn ×_k V) ↔ ({y} ∈ 𝒫₁𝒫 Nn ∧ n ∈ V))
16	1, 15	mpbiran2 885	. . . . . . . . . . . . . 14 ⊢ (⟪{y}, n⟫ ∈ (𝒫₁𝒫 Nn ×_k V) ↔ {y} ∈ 𝒫₁𝒫 Nn )
17		snelpw1 4147	. . . . . . . . . . . . . 14 ⊢ ({y} ∈ 𝒫₁𝒫 Nn ↔ y ∈ 𝒫 Nn )
18	16, 17	bitri 240	. . . . . . . . . . . . 13 ⊢ (⟪{y}, n⟫ ∈ (𝒫₁𝒫 Nn ×_k V) ↔ y ∈ 𝒫 Nn )
19	10	elpw 3729	. . . . . . . . . . . . 13 ⊢ (y ∈ 𝒫 Nn ↔ y ⊆ Nn )
20	14, 18, 19	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟪{{y}}, {n}⟫ ∈ SI_k (𝒫₁𝒫 Nn ×_k V) ↔ y ⊆ Nn )
21		snex 4112	. . . . . . . . . . . . . . . . 17 ⊢ {{z}} ∈ V
22		opkeq1 4060	. . . . . . . . . . . . . . . . . 18 ⊢ (t = {{z}} → ⟪t, ⟪{{y}}, {n}⟫⟫ = ⟪{{z}}, ⟪{{y}}, {n}⟫⟫)
23	22	eleq1d 2419	. . . . . . . . . . . . . . . . 17 ⊢ (t = {{z}} → (⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ↔ ⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
24	21, 23	ceqsexv 2895	. . . . . . . . . . . . . . . 16 ⊢ (∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ ⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)))
25		elin 3220	. . . . . . . . . . . . . . . 16 ⊢ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ↔ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∧ ⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)))
26		vex 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ z ∈ V
27		snex 4112	. . . . . . . . . . . . . . . . . . 19 ⊢ {n} ∈ V
28	26, 3, 27	otkelins2k 4256	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ ⟪z, {n}⟫ ∈ ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
29	26, 27	opkelcnvk 4251	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪z, {n}⟫ ∈ ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ ⟪{n}, z⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
30	1, 26	eqtfinrelk 4487	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪{n}, z⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ z = T_fin n)
31	28, 29, 30	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ z = T_fin n)
32		opkex 4114	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟪z, {{y}}⟫ ∈ V
33	32	elimak 4260	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟪z, {{y}}⟫ ∈ (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ))
34		elpw121c 4149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (t ∈ 𝒫₁𝒫₁1_c ↔ ∃w t = {{{w}}})
35	34	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ (∃w t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
36		19.41v 1901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃w(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ (∃w t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
37	35, 36	bitr4i 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ ∃w(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
38	37	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ ∃t∃w(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
39		df-rex 2621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) ↔ ∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
40		excom 1741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃w∃t(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ ∃t∃w(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
41	38, 39, 40	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) ↔ ∃w∃t(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
42		snex 4112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {{{w}}} ∈ V
43		opkeq1 4060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (t = {{{w}}} → ⟪t, ⟪z, {{y}}⟫⟫ = ⟪{{{w}}}, ⟪z, {{y}}⟫⟫)
44	43	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = {{{w}}} → (⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) ↔ ⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )))
45	42, 44	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃t(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ ⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ))
46		elin 3220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) ↔ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∧ ⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins3_k S_k ))
47		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {w} ∈ V
48	47, 26, 3	otkelins2k 4256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ⟪{w}, {{y}}⟫ ∈ SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))
49		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ w ∈ V
50	49, 8	opksnelsik 4266	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟪{w}, {{y}}⟫ ∈ SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ⟪w, {y}⟫ ∈ ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))
51		opkex 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ⟪w, {y}⟫ ∈ V
52	51	elimak 4260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟪w, {y}⟫ ∈ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)))
53		elpw121c 4149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (t ∈ 𝒫₁𝒫₁1_c ↔ ∃a t = {{{a}}})
54	53	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ (∃a t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
55		19.41v 1901	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃a(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ (∃a t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
56	54, 55	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ∃a(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
57	56	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ∃t∃a(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
58		df-rex 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ ∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
59		excom 1741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃a∃t(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ∃t∃a(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
60	57, 58, 59	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ ∃a∃t(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
61		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {{{a}}} ∈ V
62		opkeq1 4060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (t = {{{a}}} → ⟪t, ⟪w, {y}⟫⟫ = ⟪{{{a}}}, ⟪w, {y}⟫⟫)
63	62	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (t = {{{a}}} → (⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ ⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))))
64	61, 63	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃t(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)))
65		elsymdif 3224	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ ¬ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins3_k S_k ↔ ⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)))
66		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ {a} ∈ V
67	66, 49, 8	otkelins3k 4257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins3_k S_k ↔ ⟪{a}, w⟫ ∈ S_k )
68		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ a ∈ V
69	68, 49	elssetk 4271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟪{a}, w⟫ ∈ S_k ↔ a ∈ w)
70	67, 69	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins3_k S_k ↔ a ∈ w)
71	66, 49, 8	otkelins2k 4256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ⟪{a}, {y}⟫ ∈ SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))
72		opkex 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ⟪a, y⟫ ∈ V
73	72	elimak 4260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟪a, y⟫ ∈ (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))))
74		elpw121c 4149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (t ∈ 𝒫₁𝒫₁1_c ↔ ∃x t = {{{x}}})
75	74	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
76		19.41v 1901	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (∃x(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
77	75, 76	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ ∃x(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
78	77	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
79		df-rex 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ↔ ∃t(t ∈ 𝒫₁𝒫₁1_c ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
80		excom 1741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
81	78, 79, 80	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∃t ∈ 𝒫₁ 𝒫₁1_c⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ↔ ∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
82		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ {{{x}}} ∈ V
83		opkeq1 4060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (t = {{{x}}} → ⟪t, ⟪a, y⟫⟫ = ⟪{{{x}}}, ⟪a, y⟫⟫)
84	83	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (t = {{{x}}} → (⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ↔ ⟪{{{x}}}, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))))
85	82, 84	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ ⟪{{{x}}}, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))))
86		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ↔ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins2_k S_k ∧ ⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))))
87		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ {x} ∈ V
88	87, 68, 10	otkelins2k 4256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins2_k S_k ↔ ⟪{x}, y⟫ ∈ S_k )
89		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ x ∈ V
90	89, 10	elssetk 4271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟪{x}, y⟫ ∈ S_k ↔ x ∈ y)
91	88, 90	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins2_k S_k ↔ x ∈ y)
92	87, 68, 10	otkelins3k 4257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ ⟪{x}, a⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
93	89, 68	eqtfinrelk 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟪{x}, a⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ a = T_fin x)
94	92, 93	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ a = T_fin x)
95	91, 94	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins2_k S_k ∧ ⟪{{{x}}}, ⟪a, y⟫⟫ ∈ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ↔ (x ∈ y ∧ a = T_fin x))
96	85, 86, 95	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ (x ∈ y ∧ a = T_fin x))
97	96	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪a, y⟫⟫ ∈ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))) ↔ ∃x(x ∈ y ∧ a = T_fin x))
98	73, 81, 97	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟪a, y⟫ ∈ (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ∃x(x ∈ y ∧ a = T_fin x))
99	68, 10	opksnelsik 4266	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟪{a}, {y}⟫ ∈ SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ⟪a, y⟫ ∈ (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))
100		df-rex 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∃x ∈ y a = T_fin x ↔ ∃x(x ∈ y ∧ a = T_fin x))
101	98, 99, 100	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟪{a}, {y}⟫ ∈ SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ∃x ∈ y a = T_fin x)
102	71, 101	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ↔ ∃x ∈ y a = T_fin x)
103	70, 102	bibi12i 306	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins3_k S_k ↔ ⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
104	103	notbii 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins3_k S_k ↔ ⟪{{{a}}}, ⟪w, {y}⟫⟫ ∈ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ↔ ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
105	64, 65, 104	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃t(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
106	105	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃a∃t(t = {{{a}}} ∧ ⟪t, ⟪w, {y}⟫⟫ ∈ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c))) ↔ ∃a ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
107	52, 60, 106	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟪w, {y}⟫ ∈ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ∃a ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
108	107	notbii 287	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ ⟪w, {y}⟫ ∈ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ¬ ∃a ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
109	51	elcompl 3226	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟪w, {y}⟫ ∈ ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ ¬ ⟪w, {y}⟫ ∈ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))
110		eqabb 2459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (w = {a ∣ ∃x ∈ y a = T_fin x} ↔ ∀a(a ∈ w ↔ ∃x ∈ y a = T_fin x))
111		alex 1572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀a(a ∈ w ↔ ∃x ∈ y a = T_fin x) ↔ ¬ ∃a ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
112	110, 111	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (w = {a ∣ ∃x ∈ y a = T_fin x} ↔ ¬ ∃a ¬ (a ∈ w ↔ ∃x ∈ y a = T_fin x))
113	108, 109, 112	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟪w, {y}⟫ ∈ ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ w = {a ∣ ∃x ∈ y a = T_fin x})
114	48, 50, 113	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ↔ w = {a ∣ ∃x ∈ y a = T_fin x})
115	47, 26, 3	otkelins3k 4257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins3_k S_k ↔ ⟪{w}, z⟫ ∈ S_k )
116	49, 26	elssetk 4271	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟪{w}, z⟫ ∈ S_k ↔ w ∈ z)
117	115, 116	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins3_k S_k ↔ w ∈ z)
118	114, 117	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∧ ⟪{{{w}}}, ⟪z, {{y}}⟫⟫ ∈ Ins3_k S_k ) ↔ (w = {a ∣ ∃x ∈ y a = T_fin x} ∧ w ∈ z))
119	45, 46, 118	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃t(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ (w = {a ∣ ∃x ∈ y a = T_fin x} ∧ w ∈ z))
120	119	exbii 1582	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃w∃t(t = {{{w}}} ∧ ⟪t, ⟪z, {{y}}⟫⟫ ∈ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k )) ↔ ∃w(w = {a ∣ ∃x ∈ y a = T_fin x} ∧ w ∈ z))
121	33, 41, 120	3bitri 262	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪z, {{y}}⟫ ∈ (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ↔ ∃w(w = {a ∣ ∃x ∈ y a = T_fin x} ∧ w ∈ z))
122	26, 3, 27	otkelins3k 4257	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ↔ ⟪z, {{y}}⟫ ∈ (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))
123		df-clel 2349	. . . . . . . . . . . . . . . . . 18 ⊢ ({a ∣ ∃x ∈ y a = T_fin x} ∈ z ↔ ∃w(w = {a ∣ ∃x ∈ y a = T_fin x} ∧ w ∈ z))
124	121, 122, 123	3bitr4i 268	. . . . . . . . . . . . . . . . 17 ⊢ (⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ↔ {a ∣ ∃x ∈ y a = T_fin x} ∈ z)
125	31, 124	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ ((⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∧ ⟪{{z}}, ⟪{{y}}, {n}⟫⟫ ∈ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ↔ (z = T_fin n ∧ {a ∣ ∃x ∈ y a = T_fin x} ∈ z))
126	24, 25, 125	3bitri 262	. . . . . . . . . . . . . . 15 ⊢ (∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ (z = T_fin n ∧ {a ∣ ∃x ∈ y a = T_fin x} ∈ z))
127	126	exbii 1582	. . . . . . . . . . . . . 14 ⊢ (∃z∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ ∃z(z = T_fin n ∧ {a ∣ ∃x ∈ y a = T_fin x} ∈ z))
128		opkex 4114	. . . . . . . . . . . . . . . 16 ⊢ ⟪{{y}}, {n}⟫ ∈ V
129	128	elimak 4260	. . . . . . . . . . . . . . 15 ⊢ (⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 1_c⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)))
130		elpw11c 4148	. . . . . . . . . . . . . . . . . . 19 ⊢ (t ∈ 𝒫₁1_c ↔ ∃z t = {{z}})
131	130	anbi1i 676	. . . . . . . . . . . . . . . . . 18 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ (∃z t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
132		19.41v 1901	. . . . . . . . . . . . . . . . . 18 ⊢ (∃z(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ (∃z t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
133	131, 132	bitr4i 243	. . . . . . . . . . . . . . . . 17 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ ∃z(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
134	133	exbii 1582	. . . . . . . . . . . . . . . 16 ⊢ (∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ ∃t∃z(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
135		df-rex 2621	. . . . . . . . . . . . . . . 16 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ↔ ∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
136		excom 1741	. . . . . . . . . . . . . . . 16 ⊢ (∃z∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))) ↔ ∃t∃z(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
137	134, 135, 136	3bitr4i 268	. . . . . . . . . . . . . . 15 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ↔ ∃z∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
138	129, 137	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ∃z∃t(t = {{z}} ∧ ⟪t, ⟪{{y}}, {n}⟫⟫ ∈ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c))))
139		tfinex 4486	. . . . . . . . . . . . . . 15 ⊢ T_fin n ∈ V
140	139	clel3 2978	. . . . . . . . . . . . . 14 ⊢ ({a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n ↔ ∃z(z = T_fin n ∧ {a ∣ ∃x ∈ y a = T_fin x} ∈ z))
141	127, 138, 140	3bitr4i 268	. . . . . . . . . . . . 13 ⊢ (⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c) ↔ {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)
142	141	notbii 287	. . . . . . . . . . . 12 ⊢ (¬ ⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ¬ {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)
143	20, 142	anbi12i 678	. . . . . . . . . . 11 ⊢ ((⟪{{y}}, {n}⟫ ∈ SI_k (𝒫₁𝒫 Nn ×_k V) ∧ ¬ ⟪{{y}}, {n}⟫ ∈ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ (y ⊆ Nn ∧ ¬ {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
144		annim 414	. . . . . . . . . . 11 ⊢ ((y ⊆ Nn ∧ ¬ {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n) ↔ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
145	13, 143, 144	3bitri 262	. . . . . . . . . 10 ⊢ (⟪{{y}}, {n}⟫ ∈ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
146	12, 145	anbi12i 678	. . . . . . . . 9 ⊢ ((⟪{{y}}, {n}⟫ ∈ SI_k S_k ∧ ⟪{{y}}, {n}⟫ ∈ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ (y ∈ n ∧ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)))
147	6, 7, 146	3bitri 262	. . . . . . . 8 ⊢ (∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ (y ∈ n ∧ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)))
148	147	exbii 1582	. . . . . . 7 ⊢ (∃y∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ ∃y(y ∈ n ∧ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)))
149	27	elimak 4260	. . . . . . . 8 ⊢ ({n} ∈ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 1_c⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))))
150		elpw11c 4148	. . . . . . . . . . . 12 ⊢ (t ∈ 𝒫₁1_c ↔ ∃y t = {{y}})
151	150	anbi1i 676	. . . . . . . . . . 11 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ (∃y t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
152		19.41v 1901	. . . . . . . . . . 11 ⊢ (∃y(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ (∃y t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
153	151, 152	bitr4i 243	. . . . . . . . . 10 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ ∃y(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
154	153	exbii 1582	. . . . . . . . 9 ⊢ (∃t(t ∈ 𝒫₁1_c ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ ∃t∃y(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
155		df-rex 2621	. . . . . . . . 9 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃t(t ∈ 𝒫₁1_c ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
156		excom 1741	. . . . . . . . 9 ⊢ (∃y∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))) ↔ ∃t∃y(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
157	154, 155, 156	3bitr4i 268	. . . . . . . 8 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃y∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
158	149, 157	bitri 240	. . . . . . 7 ⊢ ({n} ∈ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∃y∃t(t = {{y}} ∧ ⟪t, {n}⟫ ∈ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)))))
159		df-rex 2621	. . . . . . 7 ⊢ (∃y ∈ n ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n) ↔ ∃y(y ∈ n ∧ ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)))
160	148, 158, 159	3bitr4i 268	. . . . . 6 ⊢ ({n} ∈ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∃y ∈ n ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
161	160	notbii 287	. . . . 5 ⊢ (¬ {n} ∈ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ¬ ∃y ∈ n ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
162	27	elcompl 3226	. . . . 5 ⊢ ({n} ∈ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ¬ {n} ∈ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c))
163		dfral2 2627	. . . . 5 ⊢ (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n) ↔ ¬ ∃y ∈ n ¬ (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
164	161, 162, 163	3bitr4i 268	. . . 4 ⊢ ({n} ∈ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
165	2, 164	bitri 240	. . 3 ⊢ (n ∈ ⋃₁ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n))
166	165	eqabi 2465	. 2 ⊢ ⋃₁ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) = {n ∣ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)}
167		ssetkex 4295	. . . . . . 7 ⊢ S_k ∈ V
168	167	sikex 4298	. . . . . 6 ⊢ SI_k S_k ∈ V
169		nncex 4397	. . . . . . . . . . 11 ⊢ Nn ∈ V
170	169	pwex 4330	. . . . . . . . . 10 ⊢ 𝒫 Nn ∈ V
171	170	pw1ex 4304	. . . . . . . . 9 ⊢ 𝒫₁𝒫 Nn ∈ V
172		vvex 4110	. . . . . . . . 9 ⊢ V ∈ V
173	171, 172	xpkex 4290	. . . . . . . 8 ⊢ (𝒫₁𝒫 Nn ×_k V) ∈ V
174	173	sikex 4298	. . . . . . 7 ⊢ SI_k (𝒫₁𝒫 Nn ×_k V) ∈ V
175		tfinrelkex 4488	. . . . . . . . . . 11 ⊢ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∈ V
176	175	cnvkex 4288	. . . . . . . . . 10 ⊢ ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∈ V
177	176	ins2kex 4308	. . . . . . . . 9 ⊢ Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∈ V
178	167	ins3kex 4309	. . . . . . . . . . . . . . . . 17 ⊢ Ins3_k S_k ∈ V
179	167	ins2kex 4308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ins2_k S_k ∈ V
180	175	ins3kex 4309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∈ V
181	179, 180	inex 4106	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) ∈ V
182		1cex 4143	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1_c ∈ V
183	182	pw1ex 4304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫₁1_c ∈ V
184	183	pw1ex 4304	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝒫₁𝒫₁1_c ∈ V
185	181, 184	imakex 4301	. . . . . . . . . . . . . . . . . . 19 ⊢ (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ∈ V
186	185	sikex 4298	. . . . . . . . . . . . . . . . . 18 ⊢ SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ∈ V
187	186	ins2kex 4308	. . . . . . . . . . . . . . . . 17 ⊢ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c) ∈ V
188	178, 187	symdifex 4109	. . . . . . . . . . . . . . . 16 ⊢ ( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) ∈ V
189	188, 184	imakex 4301	. . . . . . . . . . . . . . 15 ⊢ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
190	189	complex 4105	. . . . . . . . . . . . . 14 ⊢ ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
191	190	sikex 4298	. . . . . . . . . . . . 13 ⊢ SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
192	191	ins2kex 4308	. . . . . . . . . . . 12 ⊢ Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
193	192, 178	inex 4106	. . . . . . . . . . 11 ⊢ ( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) ∈ V
194	193, 184	imakex 4301	. . . . . . . . . 10 ⊢ (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ∈ V
195	194	ins3kex 4309	. . . . . . . . 9 ⊢ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c) ∈ V
196	177, 195	inex 4106	. . . . . . . 8 ⊢ ( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) ∈ V
197	196, 183	imakex 4301	. . . . . . 7 ⊢ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c) ∈ V
198	174, 197	difex 4108	. . . . . 6 ⊢ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c)) ∈ V
199	168, 198	inex 4106	. . . . 5 ⊢ ( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) ∈ V
200	199, 183	imakex 4301	. . . 4 ⊢ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ∈ V
201	200	complex 4105	. . 3 ⊢ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ∈ V
202	201	uni1ex 4294	. 2 ⊢ ⋃₁ ∼ (( SI_k S_k ∩ ( SI_k (𝒫₁𝒫 Nn ×_k V) ∖ (( Ins2_k ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ∩ Ins3_k (( Ins2_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k (( Ins2_k S_k ∩ Ins3_k (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V)))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ Ins3_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁1_c))) “_k 𝒫₁1_c) ∈ V
203	166, 202	eqeltrri 2424	1 ⊢ {n ∣ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = T_fin x} ∈ T_fin n)} ∈ V

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ∀wral 2615 ∃wrex 2616 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 ⊆ wss 3258 ∅c0 3551 𝒫 cpw 3723 {csn 3738 ⟪copk 4058 ⋃₁cuni1 4134 1_cc1c 4135 𝒫₁cpw1 4136 ×_k cxpk 4175 ^◡_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 SI_k csik 4182 S_k cssetk 4184 I_k cidk 4185 Nn cnnc 4374 T_fin ctfin 4436

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088

This proof depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-addc 4379 df-nnc 4380 df-tfin 4444

This theorem is used by: tfinnn 4535

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