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Theorem cnvsng 6074
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) (Proof shortened by BJ, 12-Feb-2022.)
Assertion
Ref Expression
cnvsng ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Proof of Theorem cnvsng
StepHypRef Expression
1 cnvcnvsn 6070 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
2 relsnopg 5670 . . . 4 ((𝐵𝑊𝐴𝑉) → Rel {⟨𝐵, 𝐴⟩})
32ancoms 461 . . 3 ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐵, 𝐴⟩})
4 dfrel2 6040 . . 3 (Rel {⟨𝐵, 𝐴⟩} ↔ {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
53, 4sylib 220 . 2 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
61, 5syl5eqr 2870 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {csn 4560  cop 4566  ccnv 5548  Rel wrel 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557
This theorem is referenced by:  cnvsn  6077  opswap  6080  funsng  6399  f1oprswap  6652  cnvprop  30426
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