Theorem unexb 4532

Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)

Assertion

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

Proof of Theorem unexb

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

Step	Hyp	Ref	Expression
1		uneq1 3351	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2298	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3352	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2298	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2802	. . . 4 |- x e. _V
6		vex 2802	. . . 4 |- y e. _V
7	5, 6	unex 4531	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2865	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3367	. . . 4 |- A C_ ( A u. B )
10		ssexg 4222	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
11	9, 10	mpan 424	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
12		ssun2 3368	. . . 4 |- B C_ ( A u. B )
13		ssexg 4222	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
14	12, 13	mpan 424	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
15	11, 14	jca 306	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
16	8, 15	impbii 126	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3888

This theorem is referenced by:  unexg  4533  sucexb  4588  frecabex  6542  djuexb  7207