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Theorem opswapg 4732
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 4731 . . 3 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
21unieqd 3564 . 2 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
3 elex 2542 . . . 4 (B 𝑊B V)
4 elex 2542 . . . 4 (A 𝑉A V)
5 opexgOLD 3938 . . . 4 ((B V A V) → ⟨B, A V)
63, 4, 5syl2anr 274 . . 3 ((A 𝑉 B 𝑊) → ⟨B, A V)
7 unisng 3570 . . 3 (⟨B, A V → {⟨B, A⟩} = ⟨B, A⟩)
86, 7syl 14 . 2 ((A 𝑉 B 𝑊) → {⟨B, A⟩} = ⟨B, A⟩)
92, 8eqtrd 2055 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1375  Vcvv 2534  {csn 3349  cop 3352   cuni 3553  ccnv 4269
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-xp 4276  df-rel 4277  df-cnv 4278
This theorem is referenced by:  2nd1st  5726  cnvf1olem  5765  brtposg  5788  dftpos4  5797  tpostpos  5798
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