Theorem opkthg 4131

Description: Two Kuratowski ordered pairs are equal iff their components are equal. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkthg	⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C ∧ B = D)))

Proof of Theorem opkthg

Step	Hyp	Ref	Expression
1		simp1 955	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → A ∈ V)
2		opkth1g 4130	. . . . 5 ⊢ ((A ∈ V ∧ ⟪A, B⟫ = ⟪C, D⟫) → A = C)
3	1, 2	sylan 457	. . . 4 ⊢ (((A ∈ V ∧ B ∈ W ∧ D ∈ T) ∧ ⟪A, B⟫ = ⟪C, D⟫) → A = C)
4		simp2 956	. . . . . 6 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → B ∈ W)
5		simp3 957	. . . . . 6 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → D ∈ T)
6	4, 5	jca 518	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → (B ∈ W ∧ D ∈ T))
7		opkeq1 4059	. . . . . . . . . . 11 ⊢ (A = C → ⟪A, B⟫ = ⟪C, B⟫)
8	7	eqeq1d 2361	. . . . . . . . . 10 ⊢ (A = C → (⟪A, B⟫ = ⟪C, D⟫ ↔ ⟪C, B⟫ = ⟪C, D⟫))
9	8	biimpd 198	. . . . . . . . 9 ⊢ (A = C → (⟪A, B⟫ = ⟪C, D⟫ → ⟪C, B⟫ = ⟪C, D⟫))
10	9	impcom 419	. . . . . . . 8 ⊢ ((⟪A, B⟫ = ⟪C, D⟫ ∧ A = C) → ⟪C, B⟫ = ⟪C, D⟫)
11		df-opk 4058	. . . . . . . . . . 11 ⊢ ⟪C, B⟫ = {{C}, {C, B}}
12		df-opk 4058	. . . . . . . . . . 11 ⊢ ⟪C, D⟫ = {{C}, {C, D}}
13	11, 12	eqeq12i 2366	. . . . . . . . . 10 ⊢ (⟪C, B⟫ = ⟪C, D⟫ ↔ {{C}, {C, B}} = {{C}, {C, D}})
14		prex 4112	. . . . . . . . . . 11 ⊢ {C, B} ∈ V
15		prex 4112	. . . . . . . . . . 11 ⊢ {C, D} ∈ V
16	14, 15	preqr2 4125	. . . . . . . . . 10 ⊢ ({{C}, {C, B}} = {{C}, {C, D}} → {C, B} = {C, D})
17	13, 16	sylbi 187	. . . . . . . . 9 ⊢ (⟪C, B⟫ = ⟪C, D⟫ → {C, B} = {C, D})
18		preqr2g 4126	. . . . . . . . 9 ⊢ ((B ∈ W ∧ D ∈ T) → ({C, B} = {C, D} → B = D))
19	17, 18	syl5 28	. . . . . . . 8 ⊢ ((B ∈ W ∧ D ∈ T) → (⟪C, B⟫ = ⟪C, D⟫ → B = D))
20	10, 19	syl5 28	. . . . . . 7 ⊢ ((B ∈ W ∧ D ∈ T) → ((⟪A, B⟫ = ⟪C, D⟫ ∧ A = C) → B = D))
21	20	exp3a 425	. . . . . 6 ⊢ ((B ∈ W ∧ D ∈ T) → (⟪A, B⟫ = ⟪C, D⟫ → (A = C → B = D)))
22	21	imp 418	. . . . 5 ⊢ (((B ∈ W ∧ D ∈ T) ∧ ⟪A, B⟫ = ⟪C, D⟫) → (A = C → B = D))
23	6, 22	sylan 457	. . . 4 ⊢ (((A ∈ V ∧ B ∈ W ∧ D ∈ T) ∧ ⟪A, B⟫ = ⟪C, D⟫) → (A = C → B = D))
24	3, 23	jcai 522	. . 3 ⊢ (((A ∈ V ∧ B ∈ W ∧ D ∈ T) ∧ ⟪A, B⟫ = ⟪C, D⟫) → (A = C ∧ B = D))
25	24	ex 423	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → (⟪A, B⟫ = ⟪C, D⟫ → (A = C ∧ B = D)))
26		opkeq12 4061	. 2 ⊢ ((A = C ∧ B = D) → ⟪A, B⟫ = ⟪C, D⟫)
27	25, 26	impbid1 194	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ D ∈ T) → (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C ∧ B = D)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  {csn 3737  {cpr 3738  ⟪copk 4057

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-sn 4087

This theorem 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058

This theorem is referenced by:  opkth  4132  opkelopkabg  4245  otkelins2kg  4253  otkelins3kg  4254  opkelcokg  4261