Theorem prel12 1875

Description: Equality of two unordered pairs.

Hypotheses

Ref	Expression
preq12b.1	⊢ A ∈ V
preq12b.2	⊢ B ∈ V
preq12b.3	⊢ C ∈ V
preq12b.4	⊢ D ∈ V

Assertion

Ref	Expression
prel12	⊢ (¬ A = B → ({A, B} = {C, D} ↔ (A ∈ {C, D} ∧ B ∈ {C, D})))

Proof of Theorem prel12

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . . 6 ⊢ A ∈ V
2	1	pri1 1841	. . . . 5 ⊢ A ∈ {A, B}
3		eleq2 1150	. . . . 5 ⊢ ({A, B} = {C, D} → (A ∈ {A, B} ↔ A ∈ {C, D}))
4	2, 3	mpbii 168	. . . 4 ⊢ ({A, B} = {C, D} → A ∈ {C, D})
5		preq12b.2	. . . . . 6 ⊢ B ∈ V
6	5	pri2 1842	. . . . 5 ⊢ B ∈ {A, B}
7		eleq2 1150	. . . . 5 ⊢ ({A, B} = {C, D} → (B ∈ {A, B} ↔ B ∈ {C, D}))
8	6, 7	mpbii 168	. . . 4 ⊢ ({A, B} = {C, D} → B ∈ {C, D})
9	4, 8	jca 236	. . 3 ⊢ ({A, B} = {C, D} → (A ∈ {C, D} ∧ B ∈ {C, D}))
10	9	a1i 7	. 2 ⊢ (¬ A = B → ({A, B} = {C, D} → (A ∈ {C, D} ∧ B ∈ {C, D})))
11		cleq2 1110	. . . . . . . . . . . 12 ⊢ (B = D → (A = B ↔ A = D))
12	11	negbid 463	. . . . . . . . . . 11 ⊢ (B = D → (¬ A = B ↔ ¬ A = D))
13		orel2 213	. . . . . . . . . . 11 ⊢ (¬ A = D → ((A = C ∨ A = D) → A = C))
14	12, 13	syl6bi 187	. . . . . . . . . 10 ⊢ (B = D → (¬ A = B → ((A = C ∨ A = D) → A = C)))
15	14	com3l 34	. . . . . . . . 9 ⊢ (¬ A = B → ((A = C ∨ A = D) → (B = D → A = C)))
16	15	imp 277	. . . . . . . 8 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → (B = D → A = C))
17	16	ancrd 247	. . . . . . 7 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → (B = D → (A = C ∧ B = D)))
18		cleq2 1110	. . . . . . . . . . . 12 ⊢ (B = C → (A = B ↔ A = C))
19	18	negbid 463	. . . . . . . . . . 11 ⊢ (B = C → (¬ A = B ↔ ¬ A = C))
20		orel1 212	. . . . . . . . . . 11 ⊢ (¬ A = C → ((A = C ∨ A = D) → A = D))
21	19, 20	syl6bi 187	. . . . . . . . . 10 ⊢ (B = C → (¬ A = B → ((A = C ∨ A = D) → A = D)))
22	21	com3l 34	. . . . . . . . 9 ⊢ (¬ A = B → ((A = C ∨ A = D) → (B = C → A = D)))
23	22	imp 277	. . . . . . . 8 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → (B = C → A = D))
24	23	ancrd 247	. . . . . . 7 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → (B = C → (A = D ∧ B = C)))
25	17, 24	orim12d 436	. . . . . 6 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → ((B = D ∨ B = C) → ((A = C ∧ B = D) ∨ (A = D ∧ B = C))))
26	5	elpr 1823	. . . . . . 7 ⊢ (B ∈ {C, D} ↔ (B = C ∨ B = D))
27		orcom 209	. . . . . . 7 ⊢ ((B = C ∨ B = D) ↔ (B = D ∨ B = C))
28	26, 27	bitr 151	. . . . . 6 ⊢ (B ∈ {C, D} ↔ (B = D ∨ B = C))
29		preq12b.3	. . . . . . 7 ⊢ C ∈ V
30		preq12b.4	. . . . . . 7 ⊢ D ∈ V
31	1, 5, 29, 30	preq12b 1874	. . . . . 6 ⊢ ({A, B} = {C, D} ↔ ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))
32	25, 28, 31	3imtr4g 426	. . . . 5 ⊢ ((¬ A = B ∧ (A = C ∨ A = D)) → (B ∈ {C, D} → {A, B} = {C, D}))
33	32	exp 291	. . . 4 ⊢ (¬ A = B → ((A = C ∨ A = D) → (B ∈ {C, D} → {A, B} = {C, D})))
34	1	elpr 1823	. . . 4 ⊢ (A ∈ {C, D} ↔ (A = C ∨ A = D))
35	33, 34	syl5ib 181	. . 3 ⊢ (¬ A = B → (A ∈ {C, D} → (B ∈ {C, D} → {A, B} = {C, D})))
36	35	imp3a 279	. 2 ⊢ (¬ A = B → ((A ∈ {C, D} ∧ B ∈ {C, D}) → {A, B} = {C, D}))
37	10, 36	impbid 397	1 ⊢ (¬ A = B → ({A, B} = {C, D} ↔ (A ∈ {C, D} ∧ B ∈ {C, D})))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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