Theorem cnvsn 6192

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by BJ, 12-Feb-2022.)

Hypotheses

Ref	Expression
cnvsn.1	⊢ 𝐴 ∈ V
cnvsn.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
cnvsn	⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Proof of Theorem cnvsn

Step	Hyp	Ref	Expression
1		cnvsn.1	. 2 ⊢ 𝐴 ∈ V
2		cnvsn.2	. 2 ⊢ 𝐵 ∈ V
3		cnvsng 6189	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
4	1, 2, 3	mp2an 693	1 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588  ^◡ccnv 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640

This theorem is referenced by:  op2ndb  6193  f1osn  6823  cnvfi  9112  ex-cnv  30524