Theorem bj-prexg 14232

Description: Proof of prexg 4205 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 3667	. . . . . 6 |- ( y = B -> { x , y } = { x , B } )
2	1	eleq1d 2244	. . . . 5 |- ( y = B -> ( { x , y } e. _V <-> { x , B } e. _V ) )
3		bj-zfpair2 14231	. . . . 5 |- { x , y } e. _V
4	2, 3	vtoclg 2795	. . . 4 |- ( B e. W -> { x , B } e. _V )
5		preq1 3666	. . . . 5 |- ( x = A -> { x , B } = { A , B } )
6	5	eleq1d 2244	. . . 4 |- ( x = A -> ( { x , B } e. _V <-> { A , B } e. _V ) )
7	4, 6	syl5ib 154	. . 3 |- ( x = A -> ( B e. W -> { A , B } e. _V ) )
8	7	vtocleg 2806	. 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 1353   e. wcel 2146   _Vcvv 2735   {cpr 3590

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-pr 4203  ax-bdor 14137  ax-bdeq 14141  ax-bdsep 14205

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596

This theorem is referenced by:  bj-snexg  14233  bj-unex  14240