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Theorem opexgOLD 3939
Description: An ordered pair of sets is a set. This is a special case of opexg 3938 and new proofs should use opexg 3938 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) TODO: remove in favor of opexg 3938.
Assertion
Ref Expression
opexgOLD ((A V B V) → ⟨A, B V)

Proof of Theorem opexgOLD
StepHypRef Expression
1 dfopg 3521 . 2 ((A V B V) → ⟨A, B⟩ = {{A}, {A, B}})
2 snexgOLD 3909 . . . . 5 (A V → {A} V)
32adantr 261 . . . 4 ((A V B V) → {A} V)
4 prexgOLD 3920 . . . 4 ((A V B V) → {A, B} V)
53, 4jca 290 . . 3 ((A V B V) → ({A} V {A, B} V))
6 prexgOLD 3920 . . 3 (({A} V {A, B} V) → {{A}, {A, B}} V)
75, 6syl 14 . 2 ((A V B V) → {{A}, {A, B}} V)
81, 7eqeltrd 2096 1 ((A V B V) → ⟨A, B V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  cop 3353
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 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  otth2  3952  opeliunxp  4322  opbrop  4346  relsnop  4371  op2ndb  4731  opswapg  4734  elxp4  4735  elxp5  4736  fvsn  5283  resfunexg  5307  fliftel  5358
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