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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 3563 . 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 3937 . . . 4 ((B V A V) → ⟨B, A V)
63, 4, 5syl2anr 274 . . 3 ((A 𝑉 B 𝑊) → ⟨B, A V)
7 unisng 3569 . . 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 1228   wcel 1374  Vcvv 2533  {csn 3348  cop 3351   cuni 3552  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-xp 4276  df-rel 4277  df-cnv 4278
This theorem is referenced by:  2nd1st  5727  cnvf1olem  5766  brtposg  5789  dftpos4  5798  tpostpos  5799
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