Theorem opcom 5470

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

Step	Hyp	Ref	Expression
1		opcom.1	. . 3 ⊢ 𝐴 ∈ V
2		opcom.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	opth 5444	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
4		eqcom 2769	. . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
5	4	anbi2i 632	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
6		anidm 572	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
7	3, 5, 6	3bitri 299	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by: (None)