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Theorem cnvsn 5536
 Description: Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
cnvsn {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Proof of Theorem cnvsn
StepHypRef Expression
1 cnvcnvsn 5530 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
2 cnvsn.2 . . . 4 𝐵 ∈ V
3 cnvsn.1 . . . 4 𝐴 ∈ V
42, 3relsnop 5147 . . 3 Rel {⟨𝐵, 𝐴⟩}
5 dfrel2 5502 . . 3 (Rel {⟨𝐵, 𝐴⟩} ↔ {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
64, 5mpbi 219 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩}
71, 6eqtr3i 2634 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ◡ccnv 5037  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  op2ndb  5537  cnvsng  5539  f1osn  6088  1sdom  8048  ex-cnv  26686
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