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Theorem cnvsn 5616
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 5610 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
2 cnvsn.2 . . . 4 𝐵 ∈ V
3 cnvsn.1 . . . 4 𝐴 ∈ V
42, 3relsnop 5222 . . 3 Rel {⟨𝐵, 𝐴⟩}
5 dfrel2 5581 . . 3 (Rel {⟨𝐵, 𝐴⟩} ↔ {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
64, 5mpbi 220 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩}
71, 6eqtr3i 2645 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  Vcvv 3198  {csn 4175  cop 4181  ccnv 5111  Rel wrel 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120
This theorem is referenced by:  op2ndb  5617  cnvsng  5619  f1osn  6174  1sdom  8160  ex-cnv  27278
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