Theorem prexg 4254

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 3742, prprc1 3740, and prprc2 3741. (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 3710	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2273	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 4253	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2832	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3709	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2273	. . . 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 2843	. 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 1372   e. wcel 2175   _Vcvv 2771   {cpr 3633

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639

This theorem is referenced by:  prelpwi  4257  opexg  4271  opi2  4276  opth  4280  opeqsn  4296  opeqpr  4297  uniop  4299  unex  4487  tpexg  4490  op1stb  4524  op1stbg  4525  onun2  4537  opthreg  4603  relop  4827  acexmidlemv  5941  en2prd  6908  pw2f1odclem  6930  pr2ne  7299  exmidonfinlem  7300  exmidaclem  7319  sup3exmid  9029  xrex  9977  2strbasg  12923  2stropg  12924  prdsex  13072  prdsval  13076  xpsfval  13151  xpsval  13155  gsumprval  13202  struct2slots2dom  15606  structiedg0val  15608  isomninnlem  15931  trilpolemlt1  15942  iswomninnlem  15950  iswomni0  15952  ismkvnnlem  15953