Theorem prexg 4141

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 3641, prprc1 3639, and prprc2 3640. (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 3609	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2209	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 4140	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2749	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3608	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2209	. . . 4 |- ( x = A -> ( { x , B } e. _V <-> { A , B } e. _V ) )
7	4, 6	syl5ib 153	. . 3 |- ( x = A -> ( B e. W -> { A , B } e. _V ) )
8	7	vtocleg 2760	. 2 |- ( A e. V -> ( B e. W -> { A , B } e. _V ) )
9	8	imp 123	1 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   {cpr 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  prelpwi  4144  opexg  4158  opi2  4163  opth  4167  opeqsn  4182  opeqpr  4183  uniop  4185  unex  4370  tpexg  4373  op1stb  4407  op1stbg  4408  onun2  4414  opthreg  4479  relop  4697  acexmidlemv  5780  pr2ne  7065  exmidonfinlem  7066  exmidaclem  7081  sup3exmid  8739  xrex  9669  2strbasg  12099  2stropg  12100  isomninnlem  13400  trilpolemlt1  13409  iswomninnlem  13417  ismkvnnlem  13419