Theorem bj-prexg 10031

Description: Proof of prexg 3947 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-prexg	|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )

Proof of Theorem bj-prexg

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		bj-zfpair2 10030	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2613	. . . 4 |- ( B e. W -> { 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. W -> { A , B } e. _V ) )
8	7	vtocleg 2624	. 2 |- ( A e. V -> ( B e. W -> { A , B } e. _V ) )
9	8	imp 115	1 |- ( ( A e. V /\ B e. W ) -> { 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-pr 3944  ax-bdor 9936  ax-bdeq 9940  ax-bdsep 10004

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:  bj-snexg  10032  bj-unex  10039