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

Theorem opcom 4077
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 4064 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵𝐵 = 𝐴))
4 eqcom 2090 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
54anbi2i 445 . 2 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
6 anidm 388 . 2 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
73, 5, 63bitri 204 1 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  cop 3449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator