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Theorem cnvsng 6243
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 6239 . 2 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
2 relsnopg 5813 . . . 4 ((𝐵𝑊𝐴𝑉) → Rel {⟨𝐵, 𝐴⟩})
32ancoms 458 . . 3 ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐵, 𝐴⟩})
4 dfrel2 6209 . . 3 (Rel {⟨𝐵, 𝐴⟩} ↔ {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
53, 4sylib 218 . 2 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
61, 5eqtr3id 2791 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {csn 4626  cop 4632  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  cnvsn  6246  opswap  6249  funsng  6617  f1oprswap  6892  cnvprop  32705
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