Theorem prexgOLD 3946

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 3480, prprc1 3478, and prprc2 3479. This is a special case of prexg 3947 and new proofs should use prexg 3947 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3947 and then remove it.

Assertion

Ref	Expression
prexgOLD	|- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )

Proof of Theorem prexgOLD

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq2 3448	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2106	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		zfpair2 3945	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2613	. . . 4 |- ( B e. _V -> { x , B } e. _V )
5		preq1 3447	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2106	. . . 4 |- ( x = A -> ( { x , B } e. _V <-> { A , B } e. _V ) )
7	4, 6	syl5ib 143	. . 3 |- ( x = A -> ( B e. _V -> { A , B } e. _V ) )
8	7	vtocleg 2624	. 2 |- ( A e. _V -> ( B e. _V -> { A , B } e. _V ) )
9	8	imp 115	1 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  prelpwi  3950  opexgOLD  3965  opi2  3970  opth  3974  opeqsn  3989  opeqpr  3990  uniop  3992  unex  4176  op1stb  4209  op1stbg  4210  opthreg  4280  relop  4486