Theorem un00 3567

Description: Two classes are both empty if and only if their union is empty. Dual of vvin 3569. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
un00	|- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 3378	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3556	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	eqtrdi 2287	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3392	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3272	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3562	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 122	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3393	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3272	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3562	. . . 4 |- ( B C_ (/) <-> B = (/) )
13	11, 12	sylib 122	. . 3 |- ( ( A u. B ) = (/) -> B = (/) )
14	8, 13	jca 306	. 2 |- ( ( A u. B ) = (/) -> ( A = (/) /\ B = (/) ) )
15	3, 14	impbii 126	1 |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   u. cun 3218   C_ wss 3220   (/)c0 3520

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-in1 623  ax-in2 624  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-ext 2220

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-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  undisj1  3582  undisj2  3583  disjpr2  3772