Theorem bj-prexg 13793

Description: Proof of prexg 4189 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 3654	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2235	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		bj-zfpair2 13792	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2786	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3653	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2235	. . . 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 2797	. 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 1343   e. wcel 2136   _Vcvv 2726   {cpr 3577

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-pr 4187  ax-bdor 13698  ax-bdeq 13702  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  bj-snexg  13794  bj-unex  13801