Theorem foundex 5915

Description: The class of all founded relationships is a set. (Contributed by SF, 19-Feb-2015.)

Assertion

Ref	Expression
foundex	⊢ Fr ∈ V

Proof of Theorem foundex

Dummy variables a r t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-found 5906	. . 3 ⊢ Fr = {⟨r, a⟩ ∣ ∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z))}
2		vex 2863	. . . . . . 7 ⊢ r ∈ V
3		vex 2863	. . . . . . 7 ⊢ a ∈ V
4	2, 3	opex 4589	. . . . . 6 ⊢ ⟨r, a⟩ ∈ V
5	4	elcompl 3226	. . . . 5 ⊢ (⟨r, a⟩ ∈ ∼ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ¬ ⟨r, a⟩ ∈ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))))
6		elrn2 4898	. . . . . . 7 ⊢ (⟨r, a⟩ ∈ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ∃x⟨x, ⟨r, a⟩⟩ ∈ ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))))
7		oteltxp 5783	. . . . . . . . 9 ⊢ (⟨x, ⟨r, a⟩⟩ ∈ ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ (⟨x, r⟩ ∈ ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ∧ ⟨x, a⟩ ∈ ( S ∩ ( ∼ {∅} × V))))
8		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
9	8, 2	opex 4589	. . . . . . . . . . . 12 ⊢ ⟨x, r⟩ ∈ V
10	9	elcompl 3226	. . . . . . . . . . 11 ⊢ (⟨x, r⟩ ∈ ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ¬ ⟨x, r⟩ ∈ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c))
11		elin 3220	. . . . . . . . . . . . . . 15 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ ( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) ↔ (⟨{z}, ⟨x, r⟩⟩ ∈ Ins3 S ∧ ⟨{z}, ⟨x, r⟩⟩ ∈ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)))
12	2	otelins3 5793	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ Ins3 S ↔ ⟨{z}, x⟩ ∈ S )
13		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ z ∈ V
14	13, 8	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{z}, x⟩ ∈ S ↔ z ∈ x)
15	12, 14	bitri 240	. . . . . . . . . . . . . . . 16 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ Ins3 S ↔ z ∈ x)
16		snex 4112	. . . . . . . . . . . . . . . . . . 19 ⊢ {z} ∈ V
17	16, 9	opex 4589	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨{z}, ⟨x, r⟩⟩ ∈ V
18	17	elcompl 3226	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ↔ ¬ ⟨{z}, ⟨x, r⟩⟩ ∈ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c))
19		elin 3220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) ↔ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins3 S ∧ ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )))
20	16	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins3 S ↔ ⟨{y}, ⟨x, r⟩⟩ ∈ Ins3 S )
21	2	otelins3 5793	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{y}, ⟨x, r⟩⟩ ∈ Ins3 S ↔ ⟨{y}, x⟩ ∈ S )
22		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ y ∈ V
23	22, 8	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{y}, x⟩ ∈ S ↔ y ∈ x)
24	20, 21, 23	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins3 S ↔ y ∈ x)
25		eldif 3222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I ) ↔ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∧ ¬ ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins3 I ))
26		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) ↔ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 S ))
27	9	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins4 SI3 I ↔ ⟨{t}, ⟨{y}, {z}⟩⟩ ∈ SI3 I )
28		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ t ∈ V
29	28, 22, 13	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{t}, ⟨{y}, {z}⟩⟩ ∈ SI3 I ↔ ⟨t, ⟨y, z⟩⟩ ∈ I )
30		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (t I ⟨y, z⟩ ↔ ⟨t, ⟨y, z⟩⟩ ∈ I )
31	22, 13	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ⟨y, z⟩ ∈ V
32	31	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (t I ⟨y, z⟩ ↔ t = ⟨y, z⟩)
33	30, 32	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨t, ⟨y, z⟩⟩ ∈ I ↔ t = ⟨y, z⟩)
34	27, 29, 33	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins4 SI3 I ↔ t = ⟨y, z⟩)
35		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ {y} ∈ V
36	35	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 S ↔ ⟨{t}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins2 S )
37	16	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{t}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{t}, ⟨x, r⟩⟩ ∈ Ins2 S )
38	8	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨{t}, ⟨x, r⟩⟩ ∈ Ins2 S ↔ ⟨{t}, r⟩ ∈ S )
39	28, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨{t}, r⟩ ∈ S ↔ t ∈ r)
40	38, 39	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{t}, ⟨x, r⟩⟩ ∈ Ins2 S ↔ t ∈ r)
41	36, 37, 40	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 S ↔ t ∈ r)
42	34, 41	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 S ) ↔ (t = ⟨y, z⟩ ∧ t ∈ r))
43	26, 42	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) ↔ (t = ⟨y, z⟩ ∧ t ∈ r))
44	43	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃t⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) ↔ ∃t(t = ⟨y, z⟩ ∧ t ∈ r))
45		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ↔ ∃t⟨{t}, ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ))
46		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (yrz ↔ ⟨y, z⟩ ∈ r)
47		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨y, z⟩ ∈ r ↔ ∃t(t = ⟨y, z⟩ ∧ t ∈ r))
48	46, 47	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (yrz ↔ ∃t(t = ⟨y, z⟩ ∧ t ∈ r))
49	44, 45, 48	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ↔ yrz)
50	9	otelins3 5793	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins3 I ↔ ⟨{y}, {z}⟩ ∈ I )
51		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({y} I {z} ↔ ⟨{y}, {z}⟩ ∈ I )
52	16	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({y} I {z} ↔ {y} = {z})
53	22	sneqb 3877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({y} = {z} ↔ y = z)
54	52, 53	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({y} I {z} ↔ y = z)
55	51, 54	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨{y}, {z}⟩ ∈ I ↔ y = z)
56	50, 55	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins3 I ↔ y = z)
57	56	notbii 287	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins3 I ↔ ¬ y = z)
58	49, 57	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∧ ¬ ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins3 I ) ↔ (yrz ∧ ¬ y = z))
59	25, 58	bitri 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I ) ↔ (yrz ∧ ¬ y = z))
60	24, 59	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ Ins2 Ins3 S ∧ ⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) ↔ (y ∈ x ∧ (yrz ∧ ¬ y = z)))
61	19, 60	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) ↔ (y ∈ x ∧ (yrz ∧ ¬ y = z)))
62	61	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃y⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) ↔ ∃y(y ∈ x ∧ (yrz ∧ ¬ y = z)))
63		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ↔ ∃y⟨{y}, ⟨{z}, ⟨x, r⟩⟩⟩ ∈ ( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )))
64		df-rex 2621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃y ∈ x (yrz ∧ ¬ y = z) ↔ ∃y(y ∈ x ∧ (yrz ∧ ¬ y = z)))
65	62, 63, 64	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ↔ ∃y ∈ x (yrz ∧ ¬ y = z))
66		rexanali 2661	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃y ∈ x (yrz ∧ ¬ y = z) ↔ ¬ ∀y ∈ x (yrz → y = z))
67	65, 66	bitri 240	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ↔ ¬ ∀y ∈ x (yrz → y = z))
68	67	con2bii 322	. . . . . . . . . . . . . . . . 17 ⊢ (∀y ∈ x (yrz → y = z) ↔ ¬ ⟨{z}, ⟨x, r⟩⟩ ∈ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c))
69	18, 68	bitr4i 243	. . . . . . . . . . . . . . . 16 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ↔ ∀y ∈ x (yrz → y = z))
70	15, 69	anbi12i 678	. . . . . . . . . . . . . . 15 ⊢ ((⟨{z}, ⟨x, r⟩⟩ ∈ Ins3 S ∧ ⟨{z}, ⟨x, r⟩⟩ ∈ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) ↔ (z ∈ x ∧ ∀y ∈ x (yrz → y = z)))
71	11, 70	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨{z}, ⟨x, r⟩⟩ ∈ ( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) ↔ (z ∈ x ∧ ∀y ∈ x (yrz → y = z)))
72	71	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃z⟨{z}, ⟨x, r⟩⟩ ∈ ( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) ↔ ∃z(z ∈ x ∧ ∀y ∈ x (yrz → y = z)))
73		elima1c 4948	. . . . . . . . . . . . 13 ⊢ (⟨x, r⟩ ∈ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ∃z⟨{z}, ⟨x, r⟩⟩ ∈ ( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)))
74		df-rex 2621	. . . . . . . . . . . . 13 ⊢ (∃z ∈ x ∀y ∈ x (yrz → y = z) ↔ ∃z(z ∈ x ∧ ∀y ∈ x (yrz → y = z)))
75	72, 73, 74	3bitr4i 268	. . . . . . . . . . . 12 ⊢ (⟨x, r⟩ ∈ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ∃z ∈ x ∀y ∈ x (yrz → y = z))
76	75	notbii 287	. . . . . . . . . . 11 ⊢ (¬ ⟨x, r⟩ ∈ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z))
77	10, 76	bitri 240	. . . . . . . . . 10 ⊢ (⟨x, r⟩ ∈ ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z))
78		elin 3220	. . . . . . . . . . 11 ⊢ (⟨x, a⟩ ∈ ( S ∩ ( ∼ {∅} × V)) ↔ (⟨x, a⟩ ∈ S ∧ ⟨x, a⟩ ∈ ( ∼ {∅} × V)))
79		df-br 4641	. . . . . . . . . . . . 13 ⊢ (x S a ↔ ⟨x, a⟩ ∈ S )
80	8, 3	brsset 4759	. . . . . . . . . . . . 13 ⊢ (x S a ↔ x ⊆ a)
81	79, 80	bitr3i 242	. . . . . . . . . . . 12 ⊢ (⟨x, a⟩ ∈ S ↔ x ⊆ a)
82		opelxp 4812	. . . . . . . . . . . . . 14 ⊢ (⟨x, a⟩ ∈ ( ∼ {∅} × V) ↔ (x ∈ ∼ {∅} ∧ a ∈ V))
83	3, 82	mpbiran2 885	. . . . . . . . . . . . 13 ⊢ (⟨x, a⟩ ∈ ( ∼ {∅} × V) ↔ x ∈ ∼ {∅})
84	8	elcompl 3226	. . . . . . . . . . . . 13 ⊢ (x ∈ ∼ {∅} ↔ ¬ x ∈ {∅})
85		elsn 3749	. . . . . . . . . . . . . 14 ⊢ (x ∈ {∅} ↔ x = ∅)
86	85	necon3bbii 2548	. . . . . . . . . . . . 13 ⊢ (¬ x ∈ {∅} ↔ x ≠ ∅)
87	83, 84, 86	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟨x, a⟩ ∈ ( ∼ {∅} × V) ↔ x ≠ ∅)
88	81, 87	anbi12i 678	. . . . . . . . . . 11 ⊢ ((⟨x, a⟩ ∈ S ∧ ⟨x, a⟩ ∈ ( ∼ {∅} × V)) ↔ (x ⊆ a ∧ x ≠ ∅))
89	78, 88	bitri 240	. . . . . . . . . 10 ⊢ (⟨x, a⟩ ∈ ( S ∩ ( ∼ {∅} × V)) ↔ (x ⊆ a ∧ x ≠ ∅))
90	77, 89	anbi12ci 679	. . . . . . . . 9 ⊢ ((⟨x, r⟩ ∈ ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ∧ ⟨x, a⟩ ∈ ( S ∩ ( ∼ {∅} × V))) ↔ ((x ⊆ a ∧ x ≠ ∅) ∧ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z)))
91	7, 90	bitri 240	. . . . . . . 8 ⊢ (⟨x, ⟨r, a⟩⟩ ∈ ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ((x ⊆ a ∧ x ≠ ∅) ∧ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z)))
92	91	exbii 1582	. . . . . . 7 ⊢ (∃x⟨x, ⟨r, a⟩⟩ ∈ ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ∃x((x ⊆ a ∧ x ≠ ∅) ∧ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z)))
93		exanali 1585	. . . . . . 7 ⊢ (∃x((x ⊆ a ∧ x ≠ ∅) ∧ ¬ ∃z ∈ x ∀y ∈ x (yrz → y = z)) ↔ ¬ ∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z)))
94	6, 92, 93	3bitri 262	. . . . . 6 ⊢ (⟨r, a⟩ ∈ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ¬ ∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z)))
95	94	con2bii 322	. . . . 5 ⊢ (∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z)) ↔ ¬ ⟨r, a⟩ ∈ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))))
96	5, 95	bitr4i 243	. . . 4 ⊢ (⟨r, a⟩ ∈ ∼ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ↔ ∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z)))
97	96	opabbi2i 4867	. . 3 ⊢ ∼ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) = {⟨r, a⟩ ∣ ∀x((x ⊆ a ∧ x ≠ ∅) → ∃z ∈ x ∀y ∈ x (yrz → y = z))}
98	1, 97	eqtr4i 2376	. 2 ⊢ Fr = ∼ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V)))
99		ssetex 4745	. . . . . . . . 9 ⊢ S ∈ V
100	99	ins3ex 5799	. . . . . . . 8 ⊢ Ins3 S ∈ V
101	100	ins2ex 5798	. . . . . . . . . . 11 ⊢ Ins2 Ins3 S ∈ V
102		idex 5505	. . . . . . . . . . . . . . . 16 ⊢ I ∈ V
103	102	si3ex 5807	. . . . . . . . . . . . . . 15 ⊢ SI3 I ∈ V
104	103	ins4ex 5800	. . . . . . . . . . . . . 14 ⊢ Ins4 SI3 I ∈ V
105	99	ins2ex 5798	. . . . . . . . . . . . . . . 16 ⊢ Ins2 S ∈ V
106	105	ins2ex 5798	. . . . . . . . . . . . . . 15 ⊢ Ins2 Ins2 S ∈ V
107	106	ins2ex 5798	. . . . . . . . . . . . . 14 ⊢ Ins2 Ins2 Ins2 S ∈ V
108	104, 107	inex 4106	. . . . . . . . . . . . 13 ⊢ ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) ∈ V
109		1cex 4143	. . . . . . . . . . . . 13 ⊢ 1c ∈ V
110	108, 109	imaex 4748	. . . . . . . . . . . 12 ⊢ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∈ V
111	102	ins3ex 5799	. . . . . . . . . . . 12 ⊢ Ins3 I ∈ V
112	110, 111	difex 4108	. . . . . . . . . . 11 ⊢ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I ) ∈ V
113	101, 112	inex 4106	. . . . . . . . . 10 ⊢ ( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) ∈ V
114	113, 109	imaex 4748	. . . . . . . . 9 ⊢ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ∈ V
115	114	complex 4105	. . . . . . . 8 ⊢ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c) ∈ V
116	100, 115	inex 4106	. . . . . . 7 ⊢ ( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) ∈ V
117	116, 109	imaex 4748	. . . . . 6 ⊢ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ∈ V
118	117	complex 4105	. . . . 5 ⊢ ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ∈ V
119		snex 4112	. . . . . . . 8 ⊢ {∅} ∈ V
120	119	complex 4105	. . . . . . 7 ⊢ ∼ {∅} ∈ V
121		vvex 4110	. . . . . . 7 ⊢ V ∈ V
122	120, 121	xpex 5116	. . . . . 6 ⊢ ( ∼ {∅} × V) ∈ V
123	99, 122	inex 4106	. . . . 5 ⊢ ( S ∩ ( ∼ {∅} × V)) ∈ V
124	118, 123	txpex 5786	. . . 4 ⊢ ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ∈ V
125	124	rnex 5108	. . 3 ⊢ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ∈ V
126	125	complex 4105	. 2 ⊢ ∼ ran ( ∼ (( Ins3 S ∩ ∼ (( Ins2 Ins3 S ∩ ((( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 S ) “ 1c) ∖ Ins3 I )) “ 1c)) “ 1c) ⊗ ( S ∩ ( ∼ {∅} × V))) ∈ V
127	98, 126	eqeltri 2423	1 ⊢ Fr ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551  {csn 3738  1_cc1c 4135  ⟨cop 4562  {copab 4623   class class class wbr 4640   S csset 4720   “ cima 4723   I cid 4764   × cxp 4771  ran crn 4774   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Fr cfound 5895

This theorem was proved from 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-13 1712  ax-14 1714  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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-found 5906

This theorem is referenced by:  weex  5920