Theorem unexb 4427

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 3274	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	eleq1d 2239	. . 3 |- ( x = A -> ( ( x u. y ) e. _V <-> ( A u. y ) e. _V ) )
3		uneq2 3275	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
4	3	eleq1d 2239	. . 3 |- ( y = B -> ( ( A u. y ) e. _V <-> ( A u. B ) e. _V ) )
5		vex 2733	. . . 4 |- x e. _V
6		vex 2733	. . . 4 |- y e. _V
7	5, 6	unex 4426	. . 3 |- ( x u. y ) e. _V
8	2, 4, 7	vtocl2g 2794	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
9		ssun1 3290	. . . 4 |- A C_ ( A u. B )
10		ssexg 4128	. . . 4 |- ( ( A C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> A e. _V )
11	9, 10	mpan 422	. . 3 |- ( ( A u. B ) e. _V -> A e. _V )
12		ssun2 3291	. . . 4 |- B C_ ( A u. B )
13		ssexg 4128	. . . 4 |- ( ( B C_ ( A u. B ) /\ ( A u. B ) e. _V ) -> B e. _V )
14	12, 13	mpan 422	. . 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 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  unexg  4428  sucexb  4481  frecabex  6377  djuexb  7021