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Theorem sfintfinlem1 4532

Description: Lemma for sfintfin 4533. Set up induction stratification. (Contributed by SF, 31-Jan-2015.)

**Assertion**
Ref	Expression
sfintfinlem1	⊢ {m ∣ ∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n))} ∈ V

Distinct variable group: m,n

Proof of Theorem sfintfinlem1 Dummy variables t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 ⊢ m ∈ V
2	1	eluni1 4174	. . . 4 ⊢ (m ∈ ⋃₁ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ {m} ∈ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c))
3		snex 4112	. . . . . . . 8 ⊢ {m} ∈ V
4	3	elimak 4260	. . . . . . 7 ⊢ ({m} ∈ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ∃t ∈ 1_c ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)))
5		el1c 4140	. . . . . . . . . . 11 ⊢ (t ∈ 1_c ↔ ∃n t = {n})
6	5	anbi1i 676	. . . . . . . . . 10 ⊢ ((t ∈ 1_c ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ (∃n t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
7		19.41v 1901	. . . . . . . . . 10 ⊢ (∃n(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ (∃n t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
8	6, 7	bitr4i 243	. . . . . . . . 9 ⊢ ((t ∈ 1_c ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃n(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
9	8	exbii 1582	. . . . . . . 8 ⊢ (∃t(t ∈ 1_c ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃t∃n(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
10		df-rex 2621	. . . . . . . 8 ⊢ (∃t ∈ 1_c ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ∃t(t ∈ 1_c ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
11		excom 1741	. . . . . . . 8 ⊢ (∃n∃t(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃t∃n(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
12	9, 10, 11	3bitr4i 268	. . . . . . 7 ⊢ (∃t ∈ 1_c ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ∃n∃t(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
13		snex 4112	. . . . . . . . . 10 ⊢ {n} ∈ V
14		opkeq1 4060	. . . . . . . . . . 11 ⊢ (t = {n} → ⟪t, {m}⟫ = ⟪{n}, {m}⟫)
15	14	eleq1d 2419	. . . . . . . . . 10 ⊢ (t = {n} → (⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ⟪{n}, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))))
16	13, 15	ceqsexv 2895	. . . . . . . . 9 ⊢ (∃t(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ⟪{n}, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)))
17	13, 3	opkelcnvk 4251	. . . . . . . . 9 ⊢ (⟪{n}, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ⟪{m}, {n}⟫ ∈ ( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)))
18		eldif 3222	. . . . . . . . . 10 ⊢ (⟪{m}, {n}⟫ ∈ ( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ (⟪{m}, {n}⟫ ∈ SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∧ ¬ ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)))
19		vex 2863	. . . . . . . . . . . . 13 ⊢ n ∈ V
20	1, 19	opksnelsik 4266	. . . . . . . . . . . 12 ⊢ (⟪{m}, {n}⟫ ∈ SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ ⟪m, n⟫ ∈ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))
21	1, 19	srelk 4525	. . . . . . . . . . . 12 ⊢ (⟪m, n⟫ ∈ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ S_fin (m, n))
22	20, 21	bitri 240	. . . . . . . . . . 11 ⊢ (⟪{m}, {n}⟫ ∈ SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ S_fin (m, n))
23		opkex 4114	. . . . . . . . . . . . . . 15 ⊢ ⟪{m}, {n}⟫ ∈ V
24	23	elimak 4260	. . . . . . . . . . . . . 14 ⊢ (⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 1_c⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)))
25		df-rex 2621	. . . . . . . . . . . . . 14 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) ↔ ∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
26		elpw11c 4148	. . . . . . . . . . . . . . . . . 18 ⊢ (t ∈ 𝒫₁1_c ↔ ∃y t = {{y}})
27	26	anbi1i 676	. . . . . . . . . . . . . . . . 17 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ (∃y t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
28		19.41v 1901	. . . . . . . . . . . . . . . . 17 ⊢ (∃y(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ (∃y t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
29	27, 28	bitr4i 243	. . . . . . . . . . . . . . . 16 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ∃y(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
30	29	exbii 1582	. . . . . . . . . . . . . . 15 ⊢ (∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ∃t∃y(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
31		excom 1741	. . . . . . . . . . . . . . 15 ⊢ (∃y∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ∃t∃y(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
32	30, 31	bitr4i 243	. . . . . . . . . . . . . 14 ⊢ (∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ∃y∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
33	24, 25, 32	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ∃y∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
34		snex 4112	. . . . . . . . . . . . . . . 16 ⊢ {{y}} ∈ V
35		opkeq1 4060	. . . . . . . . . . . . . . . . 17 ⊢ (t = {{y}} → ⟪t, ⟪{m}, {n}⟫⟫ = ⟪{{y}}, ⟪{m}, {n}⟫⟫)
36	35	eleq1d 2419	. . . . . . . . . . . . . . . 16 ⊢ (t = {{y}} → (⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) ↔ ⟪{{y}}, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))))
37	34, 36	ceqsexv 2895	. . . . . . . . . . . . . . 15 ⊢ (∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ⟪{{y}}, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)))
38		elin 3220	. . . . . . . . . . . . . . 15 ⊢ (⟪{{y}}, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) ↔ (⟪{{y}}, ⟪{m}, {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))) ∧ ⟪{{y}}, ⟪{m}, {n}⟫⟫ ∈ Ins3_k (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)))
39		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ y ∈ V
40	39, 3, 13	otkelins2k 4256	. . . . . . . . . . . . . . . . 17 ⊢ (⟪{{y}}, ⟪{m}, {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))) ↔ ⟪y, {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))))
41	39, 13	opkelcnvk 4251	. . . . . . . . . . . . . . . . 17 ⊢ (⟪y, {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}, y⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
42	19, 39	eqtfinrelk 4487	. . . . . . . . . . . . . . . . 17 ⊢ (⟪{n}, y⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ y = T_fin n)
43	40, 41, 42	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟪{{y}}, ⟪{m}, {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))) ↔ y = T_fin n)
44	39, 3, 13	otkelins3k 4257	. . . . . . . . . . . . . . . . 17 ⊢ (⟪{{y}}, ⟪{m}, {n}⟫⟫ ∈ Ins3_k (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ⟪y, {m}⟫ ∈ (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))
45		opkex 4114	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟪y, {m}⟫ ∈ V
46	45	elimak 4260	. . . . . . . . . . . . . . . . . 18 ⊢ (⟪y, {m}⟫ ∈ (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∃t ∈ 𝒫₁ 1_c⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))))
47		elpw11c 4148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t ∈ 𝒫₁1_c ↔ ∃x t = {{x}})
48	47	anbi1i 676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ (∃x t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
49		19.41v 1901	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃x(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ (∃x t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
50	48, 49	bitr4i 243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((t ∈ 𝒫₁1_c ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ ∃x(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
51	50	exbii 1582	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ ∃t∃x(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
52		df-rex 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ↔ ∃t(t ∈ 𝒫₁1_c ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
53		excom 1741	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃x∃t(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ ∃t∃x(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
54	51, 52, 53	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 ⊢ (∃t ∈ 𝒫₁ 1_c⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ↔ ∃x∃t(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
55		snex 4112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {{x}} ∈ V
56		opkeq1 4060	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = {{x}} → ⟪t, ⟪y, {m}⟫⟫ = ⟪{{x}}, ⟪y, {m}⟫⟫)
57	56	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = {{x}} → (⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ↔ ⟪{{x}}, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))))
58	55, 57	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃t(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ ⟪{{x}}, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))))
59		elin 3220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ↔ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ 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))) ∧ ⟪{{x}}, ⟪y, {m}⟫⟫ ∈ Ins3_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))))
60		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ x ∈ V
61	60, 39, 3	otkelins2k 4256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ 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))) ↔ ⟪x, {m}⟫ ∈ ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
62	60, 3	opkelcnvk 4251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪x, {m}⟫ ∈ ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))) ↔ ⟪{m}, x⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ⊕ Ins3_k I_k ) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∖ ({{∅}} ×_k V))))
63	1, 60	eqtfinrelk 4487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪{m}, x⟫ ∈ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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_fin m)
64	61, 62, 63	3bitri 262	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ 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))) ↔ x = T_fin m)
65	60, 39, 3	otkelins3k 4257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ Ins3_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ ⟪x, y⟫ ∈ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))
66	60, 39	srelk 4525	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟪x, y⟫ ∈ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ S_fin (x, y))
67	65, 66	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟪{{x}}, ⟪y, {m}⟫⟫ ∈ Ins3_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ↔ S_fin (x, y))
68	64, 67	anbi12i 678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟪{{x}}, ⟪y, {m}⟫⟫ ∈ 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))) ∧ ⟪{{x}}, ⟪y, {m}⟫⟫ ∈ Ins3_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ↔ (x = T_fin m ∧ S_fin (x, y)))
69	58, 59, 68	3bitri 262	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃t(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ (x = T_fin m ∧ S_fin (x, y)))
70	69	exbii 1582	. . . . . . . . . . . . . . . . . 18 ⊢ (∃x∃t(t = {{x}} ∧ ⟪t, ⟪y, {m}⟫⟫ ∈ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)))) ↔ ∃x(x = T_fin m ∧ S_fin (x, y)))
71	46, 54, 70	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟪y, {m}⟫ ∈ (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ↔ ∃x(x = T_fin m ∧ S_fin (x, y)))
72		tfinex 4486	. . . . . . . . . . . . . . . . . 18 ⊢ T_fin m ∈ V
73		sfineq1 4527	. . . . . . . . . . . . . . . . . 18 ⊢ (x = T_fin m → ( S_fin (x, y) ↔ S_fin ( T_fin m, y)))
74	72, 73	ceqsexv 2895	. . . . . . . . . . . . . . . . 17 ⊢ (∃x(x = T_fin m ∧ S_fin (x, y)) ↔ S_fin ( T_fin m, y))
75	44, 71, 74	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟪{{y}}, ⟪{m}, {n}⟫⟫ ∈ Ins3_k (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ↔ S_fin ( T_fin m, y))
76	43, 75	anbi12i 678	. . . . . . . . . . . . . . 15 ⊢ ((⟪{{y}}, ⟪{m}, {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))) ∧ ⟪{{y}}, ⟪{m}, {n}⟫⟫ ∈ Ins3_k (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) ↔ (y = T_fin n ∧ S_fin ( T_fin m, y)))
77	37, 38, 76	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ (y = T_fin n ∧ S_fin ( T_fin m, y)))
78	77	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃y∃t(t = {{y}} ∧ ⟪t, ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c))) ↔ ∃y(y = T_fin n ∧ S_fin ( T_fin m, y)))
79		tfinex 4486	. . . . . . . . . . . . . 14 ⊢ T_fin n ∈ V
80		sfineq2 4528	. . . . . . . . . . . . . 14 ⊢ (y = T_fin n → ( S_fin ( T_fin m, y) ↔ S_fin ( T_fin m, T_fin n)))
81	79, 80	ceqsexv 2895	. . . . . . . . . . . . 13 ⊢ (∃y(y = T_fin n ∧ S_fin ( T_fin m, y)) ↔ S_fin ( T_fin m, T_fin n))
82	33, 78, 81	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c) ↔ S_fin ( T_fin m, T_fin n))
83	82	notbii 287	. . . . . . . . . . 11 ⊢ (¬ ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c) ↔ ¬ S_fin ( T_fin m, T_fin n))
84	22, 83	anbi12i 678	. . . . . . . . . 10 ⊢ ((⟪{m}, {n}⟫ ∈ SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∧ ¬ ⟪{m}, {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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ( S_fin (m, n) ∧ ¬ S_fin ( T_fin m, T_fin n)))
85		annim 414	. . . . . . . . . 10 ⊢ (( S_fin (m, n) ∧ ¬ S_fin ( T_fin m, T_fin n)) ↔ ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
86	18, 84, 85	3bitri 262	. . . . . . . . 9 ⊢ (⟪{m}, {n}⟫ ∈ ( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ↔ ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
87	16, 17, 86	3bitri 262	. . . . . . . 8 ⊢ (∃t(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
88	87	exbii 1582	. . . . . . 7 ⊢ (∃n∃t(t = {n} ∧ ⟪t, {m}⟫ ∈ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c))) ↔ ∃n ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
89	4, 12, 88	3bitri 262	. . . . . 6 ⊢ ({m} ∈ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ∃n ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
90	89	notbii 287	. . . . 5 ⊢ (¬ {m} ∈ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ¬ ∃n ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
91	3	elcompl 3226	. . . . 5 ⊢ ({m} ∈ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ¬ {m} ∈ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c))
92		alex 1572	. . . . 5 ⊢ (∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n)) ↔ ¬ ∃n ¬ ( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
93	90, 91, 92	3bitr4i 268	. . . 4 ⊢ ({m} ∈ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
94	2, 93	bitri 240	. . 3 ⊢ (m ∈ ⋃₁ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ↔ ∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n)))
95	94	eqabi 2465	. 2 ⊢ ⋃₁ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) = {m ∣ ∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n))}
96		srelkex 4526	. . . . . . . 8 ⊢ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∈ V
97	96	sikex 4298	. . . . . . 7 ⊢ SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∈ V
98		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
99	98	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
100	99	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
101	96	ins3kex 4309	. . . . . . . . . . . 12 ⊢ Ins3_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∈ V
102	100, 101	inex 4106	. . . . . . . . . . 11 ⊢ ( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) ∈ V
103		1cex 4143	. . . . . . . . . . . 12 ⊢ 1_c ∈ V
104	103	pw1ex 4304	. . . . . . . . . . 11 ⊢ 𝒫₁1_c ∈ V
105	102, 104	imakex 4301	. . . . . . . . . 10 ⊢ (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ∈ V
106	105	ins3kex 4309	. . . . . . . . 9 ⊢ Ins3_k (( 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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c) ∈ V
107	100, 106	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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) ∈ V
108	107, 104	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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c) ∈ V
109	97, 108	difex 4108	. . . . . 6 ⊢ ( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ∈ V
110	109	cnvkex 4288	. . . . 5 ⊢ ^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) ∈ V
111	110, 103	imakex 4301	. . . 4 ⊢ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ∈ V
112	111	complex 4105	. . 3 ⊢ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ∈ V
113	112	uni1ex 4294	. 2 ⊢ ⋃₁ ∼ (^◡_k( SI_k (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∖ (( 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 ^◡_k(({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_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 (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((𝒫 1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁𝒫₁1_c)) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k 𝒫₁𝒫₁1_c) ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁𝒫₁1_c))) “_k 𝒫₁1_c)) “_k 𝒫₁1_c)) “_k 1_c) ∈ V
114	95, 113	eqeltrri 2424	1 ⊢ {m ∣ ∀n( S_fin (m, n) → S_fin ( T_fin m, T_fin n))} ∈ V

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 358 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2616 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 ∅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 S_fin wsfin 4439

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 df-sfin 4447

This theorem is used by: sfintfin 4533

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