Theorem un00 3455

Description: Two classes are empty iff their union is empty. (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 3271	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3442	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	eqtrdi 2215	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3285	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3166	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 147	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3448	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 121	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3286	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3166	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 147	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3448	. . . 4 |- ( B C_ (/) <-> B = (/) )
13	11, 12	sylib 121	. . 3 |- ( ( A u. B ) = (/) -> B = (/) )
14	8, 13	jca 304	. 2 |- ( ( A u. B ) = (/) -> ( A = (/) /\ B = (/) ) )
15	3, 14	impbii 125	1 |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   u. cun 3114   C_ wss 3116   (/)c0 3409

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-in1 604  ax-in2 605  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-ext 2147

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-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by:  undisj1  3466  undisj2  3467  disjpr2  3640