Theorem bdunexb 15412

Description: Bounded version of unexb 4473. (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 3306	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2262	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3307	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2262	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2763	. . . 4 |- x e. _V
6		vex 2763	. . . 4 |- y e. _V
7	5, 6	bj-unex 15411	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2824	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3322	. . . 4 |- A C_ ( A u. B )
10		bdunex.bd1	. . . . 5 |- BOUNDED A
11	10	bdssexg 15396	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
12	9, 11	mpan 424	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
13		ssun2 3323	. . . 4 |- B C_ ( A u. B )
14		bdunex.bd2	. . . . 5 |- BOUNDED B
15	14	bdssexg 15396	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
16	13, 15	mpan 424	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
17	12, 16	jca 306	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
18	8, 17	impbii 126	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   C_ wss 3153  BOUNDED wbdc 15332

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-uni 3836  df-bdc 15333

This theorem is referenced by: (None)