Theorem bdunexb 10854

Description: Bounded version of unexb 4197. (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 3120	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2148	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3121	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2148	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2605	. . . 4 |- x e. _V
6		vex 2605	. . . 4 |- y e. _V
7	5, 6	bj-unex 10853	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2663	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3136	. . . 4 |- A C_ ( A u. B )
10		bdunex.bd1	. . . . 5 |- BOUNDED A
11	10	bdssexg 10838	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
12	9, 11	mpan 415	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
13		ssun2 3137	. . . 4 |- B C_ ( A u. B )
14		bdunex.bd2	. . . . 5 |- BOUNDED B
15	14	bdssexg 10838	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
16	13, 15	mpan 415	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
17	12, 16	jca 300	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
18	8, 17	impbii 124	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   _Vcvv 2602   u. cun 2972   C_ wss 2974  BOUNDED wbdc 10774

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-pr 3966  ax-un 4190  ax-bd0 10747  ax-bdor 10750  ax-bdex 10753  ax-bdeq 10754  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-uni 3604  df-bdc 10775

This theorem is referenced by: (None)