Theorem un00 3263

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 3092	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3251	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	syl6eq 2088	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3106	. . . . 5 |- A C_ ( A u. B )
5		sseq2 2967	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 136	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3256	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 127	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3107	. . . . 5 |- B C_ ( A u. B )
10		sseq2 2967	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 136	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3256	. . . 4 |- ( B C_ (/) <-> B = (/) )
13	11, 12	sylib 127	. . 3 |- ( ( A u. B ) = (/) -> B = (/) )
14	8, 13	jca 290	. 2 |- ( ( A u. B ) = (/) -> ( A = (/) /\ B = (/) ) )
15	3, 14	impbii 117	1 |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   u. cun 2915   C_ wss 2917   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by:  undisj1  3279  undisj2  3280  disjpr2  3434