Theorem unexb 4420

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 3269	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2235	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3270	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2235	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2729	. . . 4 |- x e. _V
6		vex 2729	. . . 4 |- y e. _V
7	5, 6	unex 4419	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2790	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3285	. . . 4 |- A C_ ( A u. B )
10		ssexg 4121	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
11	9, 10	mpan 421	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
12		ssun2 3286	. . . 4 |- B C_ ( A u. B )
13		ssexg 4121	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
14	12, 13	mpan 421	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
15	11, 14	jca 304	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
16	8, 15	impbii 125	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by:  unexg  4421  sucexb  4474  frecabex  6366  djuexb  7009