Theorem bj-prexg 15847

Description: Proof of prexg 4255 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 3711	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2274	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		bj-zfpair2 15846	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2833	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3710	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2274	. . . 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 2844	. 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 1373   e. wcel 2176   _Vcvv 2772   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-pr 4253  ax-bdor 15752  ax-bdeq 15756  ax-bdsep 15820

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  bj-snexg  15848  bj-unex  15855