Theorem bdunexb 12801

Description: Bounded version of unexb 4321. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdunex.bd1	|- BOUNDED A
bdunex.bd2	|- BOUNDED B

Assertion

Ref	Expression
bdunexb	|- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Proof of Theorem bdunexb

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 3187	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2181	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3188	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2181	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2658	. . . 4 |- x e. _V
6		vex 2658	. . . 4 |- y e. _V
7	5, 6	bj-unex 12800	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2719	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3203	. . . 4 |- A C_ ( A u. B )
10		bdunex.bd1	. . . . 5 |- BOUNDED A
11	10	bdssexg 12785	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
12	9, 11	mpan 418	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
13		ssun2 3204	. . . 4 |- B C_ ( A u. B )
14		bdunex.bd2	. . . . 5 |- BOUNDED B
15	14	bdssexg 12785	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
16	13, 15	mpan 418	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
17	12, 16	jca 302	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
18	8, 17	impbii 125	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   _Vcvv 2655   u. cun 3033   C_ wss 3035  BOUNDED wbdc 12721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-pr 4089  ax-un 4313  ax-bd0 12694  ax-bdor 12697  ax-bdex 12700  ax-bdeq 12701  ax-bdel 12702  ax-bdsb 12703  ax-bdsep 12765

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-uni 3701  df-bdc 12722

This theorem is referenced by: (None)