ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opcom GIF version

Theorem opcom 4142
Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
Hypotheses
Ref Expression
opcom.1 𝐴 ∈ V
opcom.2 𝐵 ∈ V
Assertion
Ref Expression
opcom (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)

Proof of Theorem opcom
StepHypRef Expression
1 opcom.1 . . 3 𝐴 ∈ V
2 opcom.2 . . 3 𝐵 ∈ V
31, 2opth 4129 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵𝐵 = 𝐴))
4 eqcom 2119 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
54anbi2i 452 . 2 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
6 anidm 393 . 2 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
73, 5, 63bitri 205 1 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  cop 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator