Theorem prexg 4294

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 3776, prprc1 3774, and prprc2 3775. (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 3744	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2298	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 4293	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2861	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3743	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2298	. . . 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 2874	. 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 1395   e. wcel 2200   _Vcvv 2799   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prelpw  4298  prelpwi  4299  opexg  4313  opi2  4318  opth  4322  opeqsn  4338  opeqpr  4339  uniop  4341  unex  4531  tpexg  4534  op1stb  4568  op1stbg  4569  onun2  4581  opthreg  4647  relop  4871  acexmidlemv  5998  en2prd  6968  pw2f1odclem  6991  pr2ne  7361  exmidonfinlem  7367  exmidaclem  7386  sup3exmid  9100  xrex  10048  2strbasg  13148  2stropg  13149  prdsex  13297  prdsval  13301  xpsfval  13376  xpsval  13380  gsumprval  13427  struct2slots2dom  15833  structiedg0val  15835  edgstruct  15858  umgrbien  15904  upgr1edc  15915  upgr1eopdc  15917  isomninnlem  16357  trilpolemlt1  16368  iswomninnlem  16376  iswomni0  16378  ismkvnnlem  16379