Theorem preq12b 1874

Description: Equality relationship for 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
preq12b	⊢ ({A, B} = {C, D} ↔ ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))

Proof of Theorem preq12b

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	1	elpr 1823	. . . 4 ⊢ (A ∈ {C, D} ↔ (A = C ∨ A = D))
6	4, 5	sylib 173	. . 3 ⊢ ({A, B} = {C, D} → (A = C ∨ A = D))
7		preq1 1870	. . . . . . . 8 ⊢ (A = C → {A, B} = {C, B})
8	7	cleq1d 1109	. . . . . . 7 ⊢ (A = C → ({A, B} = {C, D} ↔ {C, B} = {C, D}))
9		preq12b.2	. . . . . . . 8 ⊢ B ∈ V
10		preq12b.4	. . . . . . . 8 ⊢ D ∈ V
11	9, 10	prer2 1873	. . . . . . 7 ⊢ ({C, B} = {C, D} → B = D)
12	8, 11	syl6bi 187	. . . . . 6 ⊢ (A = C → ({A, B} = {C, D} → B = D))
13	12	com12 13	. . . . 5 ⊢ ({A, B} = {C, D} → (A = C → B = D))
14	13	ancld 246	. . . 4 ⊢ ({A, B} = {C, D} → (A = C → (A = C ∧ B = D)))
15		prcom 1840	. . . . . . 7 ⊢ {C, D} = {D, C}
16	15	cleq2i 1111	. . . . . 6 ⊢ ({A, B} = {C, D} ↔ {A, B} = {D, C})
17		preq1 1870	. . . . . . . . 9 ⊢ (A = D → {A, B} = {D, B})
18	17	cleq1d 1109	. . . . . . . 8 ⊢ (A = D → ({A, B} = {D, C} ↔ {D, B} = {D, C}))
19		preq12b.3	. . . . . . . . 9 ⊢ C ∈ V
20	9, 19	prer2 1873	. . . . . . . 8 ⊢ ({D, B} = {D, C} → B = C)
21	18, 20	syl6bi 187	. . . . . . 7 ⊢ (A = D → ({A, B} = {D, C} → B = C))
22	21	com12 13	. . . . . 6 ⊢ ({A, B} = {D, C} → (A = D → B = C))
23	16, 22	sylbi 174	. . . . 5 ⊢ ({A, B} = {C, D} → (A = D → B = C))
24	23	ancld 246	. . . 4 ⊢ ({A, B} = {C, D} → (A = D → (A = D ∧ B = C)))
25	14, 24	orim12d 436	. . 3 ⊢ ({A, B} = {C, D} → ((A = C ∨ A = D) → ((A = C ∧ B = D) ∨ (A = D ∧ B = C))))
26	6, 25	mpd 46	. 2 ⊢ ({A, B} = {C, D} → ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))
27		preq2 1871	. . . 4 ⊢ (B = D → {C, B} = {C, D})
28	7, 27	sylan9eq 1144	. . 3 ⊢ ((A = C ∧ B = D) → {A, B} = {C, D})
29		prcom 1840	. . . . 5 ⊢ {D, B} = {B, D}
30	17, 29	syl6eq 1140	. . . 4 ⊢ (A = D → {A, B} = {B, D})
31		preq1 1870	. . . 4 ⊢ (B = C → {B, D} = {C, D})
32	30, 31	sylan9eq 1144	. . 3 ⊢ ((A = D ∧ B = C) → {A, B} = {C, D})
33	28, 32	jaoi 275	. 2 ⊢ (((A = C ∧ B = D) ∨ (A = D ∧ B = C)) → {A, B} = {C, D})
34	26, 33	impbi 139	1 ⊢ ({A, B} = {C, D} ↔ ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))

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

This theorem is referenced by:  prel12 1875  preleq 3454

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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