Theorem prexg 4213

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 3704, prprc1 3702, and prprc2 3703. (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.

Step	Hyp	Ref	Expression
1		preq2 3672	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2246	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 4212	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2799	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3671	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2246	. . . 4 |- ( x = A -> ( { x , B } e. _V <-> { A , B } e. _V ) )
7	4, 6	imbitrid 154	. . 3 |- ( x = A -> ( B e. W -> { A , B } e. _V ) )
8	7	vtocleg 2810	. 2 |- ( A e. V -> ( B e. W -> { A , B } e. _V ) )
9	8	imp 124	1 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   {cpr 3595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  prelpwi  4216  opexg  4230  opi2  4235  opth  4239  opeqsn  4254  opeqpr  4255  uniop  4257  unex  4443  tpexg  4446  op1stb  4480  op1stbg  4481  onun2  4491  opthreg  4557  relop  4779  acexmidlemv  5875  pr2ne  7193  exmidonfinlem  7194  exmidaclem  7209  sup3exmid  8916  xrex  9858  2strbasg  12580  2stropg  12581  prdsex  12723  xpsfval  12772  xpsval  12776  isomninnlem  14863  trilpolemlt1  14874  iswomninnlem  14882  iswomni0  14884  ismkvnnlem  14885