Theorem cnvsn 5963

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 5960	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
4	1, 2, 3	mp2an 688	1 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Syntax hints:   = wceq 1522   ∈ wcel 2081  Vcvv 3437  {csn 4476  ⟨cop 4482  ^◡ccnv 5447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-br 4967  df-opab 5029  df-xp 5454  df-rel 5455  df-cnv 5456

This theorem is referenced by:  op2ndb  5964  f1osn  6527  1sdom  8572  ex-cnv  27913