Theorem bj-prexg 9366

Description: Proof of prexg 3938 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 3439	. . . . . 6 |- (y = B -> {x , y } = {x , B })
2	1	eleq1d 2103	. . . . 5 |- (y = B -> ( {x , y } e. _V <-> {x , B } e. _V))
3		bj-zfpair2 9365	. . . . 5 |- {x , y } e. _V
4	2, 3	vtoclg 2607	. . . 4 |- (B e. W -> {x , B } e. _V)
5		preq1 3438	. . . . 5 |- (x = A -> {x , B } = {A , B })
6	5	eleq1d 2103	. . . 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 2618	. 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 1242   e. wcel 1390   _Vcvv 2551   {cpr 3368

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-pr 3935  ax-bdor 9271  ax-bdeq 9275  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  bj-snexg  9367  bj-unex  9374