Theorem unexb 4583

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 3376	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2307	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3377	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2307	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2824	. . . 4 |- x e. _V
6		vex 2824	. . . 4 |- y e. _V
7	5, 6	unex 4582	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2887	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3392	. . . 4 |- A C_ ( A u. B )
10		ssexg 4267	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
11	9, 10	mpan 428	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
12		ssun2 3393	. . . 4 |- B C_ ( A u. B )
13		ssexg 4267	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
14	12, 13	mpan 428	. . 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 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-uni 3931

This theorem is referenced by:  unexg  4584  sucexb  4639  frecabex  6659  djuexb  7374