Theorem opcom 5442

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 5416	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
4		eqcom 2746	. . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
5	4	anbi2i 629	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
6		anidm 569	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
7	3, 5, 6	3bitri 298	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562

This theorem is referenced by: (None)