New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  opkth GIF version

Theorem opkth 4132
 Description: Two Kuratowski ordered pairs are equal iff their components are equal. (Contributed by SF, 12-Jan-2015.)
Hypotheses
Ref Expression
opkth.1 A V
opkth.2 B V
opkth.3 D V
Assertion
Ref Expression
opkth (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C B = D))

Proof of Theorem opkth
StepHypRef Expression
1 opkth.1 . 2 A V
2 opkth.2 . 2 B V
3 opkth.3 . 2 D V
4 opkthg 4131 . 2 ((A V B V D V) → (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C B = D)))
51, 2, 3, 4mp3an 1277 1 (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C B = D))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟪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: (None)
 Copyright terms: Public domain W3C validator