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Theorem cnvsn 4991
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 4985 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
2 cnvsn.2 . . . 4 𝐵 ∈ V
3 cnvsn.1 . . . 4 𝐴 ∈ V
42, 3relsnop 4615 . . 3 Rel {⟨𝐵, 𝐴⟩}
5 dfrel2 4959 . . 3 (Rel {⟨𝐵, 𝐴⟩} ↔ {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
64, 5mpbi 144 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩}
71, 6eqtr3i 2140 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  Vcvv 2660  {csn 3497  cop 3500  ccnv 4508  Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  op2ndb  4992  cnvsng  4994  f1osn  5375
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