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Theorem opswapg 4701
Description: Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
Assertion
Ref Expression
opswapg ((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)

Proof of Theorem opswapg
StepHypRef Expression
1 cnvsng 4700 . . 3 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
21unieqd 3543 . 2 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
3 elex 2541 . . . 4 (B 𝑊B V)
4 elex 2541 . . . 4 (A 𝑉A V)
5 opexgOLD 3917 . . . 4 ((B V A V) → ⟨B, A V)
63, 4, 5syl2anr 274 . . 3 ((A 𝑉 B 𝑊) → ⟨B, A V)
7 unisng 3549 . . 3 (⟨B, A V → {⟨B, A⟩} = ⟨B, A⟩)
86, 7syl 14 . 2 ((A 𝑉 B 𝑊) → {⟨B, A⟩} = ⟨B, A⟩)
92, 8eqtrd 2054 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1373   wcel 1375  Vcvv 2533  {csn 3327  cop 3330   cuni 3532  ccnv 4237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-xp 4244  df-rel 4245  df-cnv 4246
This theorem is referenced by:  2nd1st  5695  cnvf1olem  5734  brtposg  5757  dftpos4  5766  tpostpos  5767
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