Theorem prexg 4271

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 3753, prprc1 3751, and prprc2 3752. (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 3721	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2276	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 4270	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2838	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3720	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2276	. . . 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 2851	. 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 1373   e. wcel 2178   _Vcvv 2776   {cpr 3644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  prelpw  4275  prelpwi  4276  opexg  4290  opi2  4295  opth  4299  opeqsn  4315  opeqpr  4316  uniop  4318  unex  4506  tpexg  4509  op1stb  4543  op1stbg  4544  onun2  4556  opthreg  4622  relop  4846  acexmidlemv  5965  en2prd  6933  pw2f1odclem  6956  pr2ne  7326  exmidonfinlem  7332  exmidaclem  7351  sup3exmid  9065  xrex  10013  2strbasg  13067  2stropg  13068  prdsex  13216  prdsval  13220  xpsfval  13295  xpsval  13299  gsumprval  13346  struct2slots2dom  15752  structiedg0val  15754  edgstruct  15775  umgrbien  15821  upgr1edc  15829  upgr1eopdc  15831  isomninnlem  16171  trilpolemlt1  16182  iswomninnlem  16190  iswomni0  16192  ismkvnnlem  16193