Theorem cnvsn 6227

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 6224	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
4	1, 2, 3	mp2an 704	1 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Syntax hints:   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589  ⟨cop 4595  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669

This theorem is referenced by:  op2ndb  6228  f1osn  6862  cnvfi  9156  ex-cnv  30788