Theorem bj-prexg 16274

Description: Proof of prexg 4295 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 3744	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2298	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		bj-zfpair2 16273	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2861	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3743	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2298	. . . 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 2874	. 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 1395   e. wcel 2200   _Vcvv 2799   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-pr 4293  ax-bdor 16179  ax-bdeq 16183  ax-bdsep 16247

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  bj-snexg  16275  bj-unex  16282