Theorem unexb 4258

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 3145	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2156	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3146	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2156	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2622	. . . 4 |- x e. _V
6		vex 2622	. . . 4 |- y e. _V
7	5, 6	unex 4257	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2683	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3161	. . . 4 |- A C_ ( A u. B )
10		ssexg 3970	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
11	9, 10	mpan 415	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
12		ssun2 3162	. . . 4 |- B C_ ( A u. B )
13		ssexg 3970	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
14	12, 13	mpan 415	. . 3 |- ( ( A u. B ) e. _V -> B e. _V )
15	11, 14	jca 300	. 2 |- ( ( A u. B ) e. _V -> ( A e. _V /\ B e. _V ) )
16	8, 15	impbii 124	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   _Vcvv 2619   u. cun 2995   C_ wss 2997

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pr 4027  ax-un 4251

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-uni 3649

This theorem is referenced by:  unexg  4259  sucexb  4304  frecabex  6145