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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
StepHypRef 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)
31, 2sylan 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)
64, 5jca 518 . . . . 5 ((A V B W D T) → (B W D T))
7 opkeq1 4059 . . . . . . . . . . 11 (A = C → ⟪A, B⟫ = ⟪C, B⟫)
87eqeq1d 2361 . . . . . . . . . 10 (A = C → (⟪A, B⟫ = ⟪C, D⟫ ↔ ⟪C, B⟫ = ⟪C, D⟫))
98biimpd 198 . . . . . . . . 9 (A = C → (⟪A, B⟫ = ⟪C, D⟫ → ⟪C, B⟫ = ⟪C, D⟫))
109impcom 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}}
1311, 12eqeq12i 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
1614, 15preqr2 4125 . . . . . . . . . 10 ({{C}, {C, B}} = {{C}, {C, D}} → {C, B} = {C, D})
1713, 16sylbi 187 . . . . . . . . 9 (⟪C, B⟫ = ⟪C, D⟫ → {C, B} = {C, D})
18 preqr2g 4126 . . . . . . . . 9 ((B W D T) → ({C, B} = {C, D} → B = D))
1917, 18syl5 28 . . . . . . . 8 ((B W D T) → (⟪C, B⟫ = ⟪C, D⟫ → B = D))
2010, 19syl5 28 . . . . . . 7 ((B W D T) → ((⟪A, B⟫ = ⟪C, D A = C) → B = D))
2120exp3a 425 . . . . . 6 ((B W D T) → (⟪A, B⟫ = ⟪C, D⟫ → (A = CB = D)))
2221imp 418 . . . . 5 (((B W D T) A, B⟫ = ⟪C, D⟫) → (A = CB = D))
236, 22sylan 457 . . . 4 (((A V B W D T) A, B⟫ = ⟪C, D⟫) → (A = CB = D))
243, 23jcai 522 . . 3 (((A V B W D T) A, B⟫ = ⟪C, D⟫) → (A = C B = D))
2524ex 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⟫)
2725, 26impbid1 194 1 ((A V B W D T) → (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C B = D)))
 Colors of variables: wff setvar class 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
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