Theorem preq12b 4128

Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

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	prid1 3828	. . . . 5 ⊢ A ∈ {A, B}
3		eleq2 2414	. . . . 5 ⊢ ({A, B} = {C, D} → (A ∈ {A, B} ↔ A ∈ {C, D}))
4	2, 3	mpbii 202	. . . 4 ⊢ ({A, B} = {C, D} → A ∈ {C, D})
5	1	elpr 3752	. . . 4 ⊢ (A ∈ {C, D} ↔ (A = C ∨ A = D))
6	4, 5	sylib 188	. . 3 ⊢ ({A, B} = {C, D} → (A = C ∨ A = D))
7		preq1 3800	. . . . . . . 8 ⊢ (A = C → {A, B} = {C, B})
8	7	eqeq1d 2361	. . . . . . 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	preqr2 4126	. . . . . . 7 ⊢ ({C, B} = {C, D} → B = D)
12	8, 11	syl6bi 219	. . . . . 6 ⊢ (A = C → ({A, B} = {C, D} → B = D))
13	12	com12 27	. . . . 5 ⊢ ({A, B} = {C, D} → (A = C → B = D))
14	13	ancld 536	. . . 4 ⊢ ({A, B} = {C, D} → (A = C → (A = C ∧ B = D)))
15		prcom 3799	. . . . . . 7 ⊢ {C, D} = {D, C}
16	15	eqeq2i 2363	. . . . . 6 ⊢ ({A, B} = {C, D} ↔ {A, B} = {D, C})
17		preq1 3800	. . . . . . . . 9 ⊢ (A = D → {A, B} = {D, B})
18	17	eqeq1d 2361	. . . . . . . 8 ⊢ (A = D → ({A, B} = {D, C} ↔ {D, B} = {D, C}))
19		preq12b.3	. . . . . . . . 9 ⊢ C ∈ V
20	9, 19	preqr2 4126	. . . . . . . 8 ⊢ ({D, B} = {D, C} → B = C)
21	18, 20	syl6bi 219	. . . . . . 7 ⊢ (A = D → ({A, B} = {D, C} → B = C))
22	21	com12 27	. . . . . 6 ⊢ ({A, B} = {D, C} → (A = D → B = C))
23	16, 22	sylbi 187	. . . . 5 ⊢ ({A, B} = {C, D} → (A = D → B = C))
24	23	ancld 536	. . . 4 ⊢ ({A, B} = {C, D} → (A = D → (A = D ∧ B = C)))
25	14, 24	orim12d 811	. . 3 ⊢ ({A, B} = {C, D} → ((A = C ∨ A = D) → ((A = C ∧ B = D) ∨ (A = D ∧ B = C))))
26	6, 25	mpd 14	. 2 ⊢ ({A, B} = {C, D} → ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))
27		preq12 3802	. . 3 ⊢ ((A = C ∧ B = D) → {A, B} = {C, D})
28		prcom 3799	. . . . 5 ⊢ {D, B} = {B, D}
29	17, 28	syl6eq 2401	. . . 4 ⊢ (A = D → {A, B} = {B, D})
30		preq1 3800	. . . 4 ⊢ (B = C → {B, D} = {C, D})
31	29, 30	sylan9eq 2405	. . 3 ⊢ ((A = D ∧ B = C) → {A, B} = {C, D})
32	27, 31	jaoi 368	. 2 ⊢ (((A = C ∧ B = D) ∨ (A = D ∧ B = C)) → {A, B} = {C, D})
33	26, 32	impbii 180	1 ⊢ ({A, B} = {C, D} ↔ ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {cpr 3739

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  preq12bg  4129