Theorem prexgOLD 3937

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 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 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 3439	. . . . . 6 |- (y = B -> {x , y } = {x , B })
2	1	eleq1d 2103	. . . . 5 |- (y = B -> ( {x , y } e. _V <-> {x , B } e. _V))
3		zfpair2 3936	. . . . 5 |- {x , y } e. _V
4	2, 3	vtoclg 2607	. . . 4 |- (B e. _V -> {x , B } e. _V)
5		preq1 3438	. . . . 5 |- (x = A -> {x , B } = {A , B })
6	5	eleq1d 2103	. . . 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 2618	. 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 1242   e. wcel 1390   _Vcvv 2551   {cpr 3368

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  prelpwi  3941  opexgOLD  3956  opi2  3961  opth  3965  opeqsn  3980  opeqpr  3981  uniop  3983  unex  4142  op1stb  4175  op1stbg  4176  opthreg  4234  relop  4429