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Theorem prexgOLD 3937
Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 and then remove it.
Assertion
Ref Expression
prexgOLD  _V  _V  { ,  }  _V

Proof of Theorem prexgOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3439 . . . . . 6  { ,  }  { ,  }
21eleq1d 2103 . . . . 5  { ,  }  _V  { ,  }  _V
3 zfpair2 3936 . . . . 5  { ,  }  _V
42, 3vtoclg 2607 . . . 4  _V  { ,  }  _V
5 preq1 3438 . . . . 5  { ,  }  { ,  }
65eleq1d 2103 . . . 4  { ,  }  _V  { ,  }  _V
74, 6syl5ib 143 . . 3  _V  { ,  }  _V
87vtocleg 2618 . 2  _V  _V  { ,  }  _V
98imp 115 1  _V  _V  { ,  }  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551   {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  prelpwi  3941  opexgOLD  3956  opi2  3961  opth  3965  opeqsn  3980  opeqpr  3981  uniop  3983  unex  4142  op1stb  4175  op1stbg  4176  opthreg  4234  relop  4429
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