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Theorem prexg 3975
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 3508, prprc1 3506, and prprc2 3507. (Contributed by Jim Kingdon, 16-Sep-2018.)
Assertion
Ref Expression
prexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )

Proof of Theorem prexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3476 . . . . . 6  |-  ( y  =  B  ->  { x ,  y }  =  { x ,  B } )
21eleq1d 2122 . . . . 5  |-  ( y  =  B  ->  ( { x ,  y }  e.  _V  <->  { x ,  B }  e.  _V ) )
3 zfpair2 3973 . . . . 5  |-  { x ,  y }  e.  _V
42, 3vtoclg 2630 . . . 4  |-  ( B  e.  W  ->  { x ,  B }  e.  _V )
5 preq1 3475 . . . . 5  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
65eleq1d 2122 . . . 4  |-  ( x  =  A  ->  ( { x ,  B }  e.  _V  <->  { A ,  B }  e.  _V ) )
74, 6syl5ib 147 . . 3  |-  ( x  =  A  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
87vtocleg 2641 . 2  |-  ( A  e.  V  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
98imp 119 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by:  opexg  3992  tpexg  4207  onun2  4244  acexmidlemv  5538  xrex  8857
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