Theorem un00 3329

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 3149	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3316	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	syl6eq 2136	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3163	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3048	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 146	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3322	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 120	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3164	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3048	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 146	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3322	. . . 4 |- ( B C_ (/) <-> B = (/) )
13	11, 12	sylib 120	. . 3 |- ( ( A u. B ) = (/) -> B = (/) )
14	8, 13	jca 300	. 2 |- ( ( A u. B ) = (/) -> ( A = (/) /\ B = (/) ) )
15	3, 14	impbii 124	1 |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1289   u. cun 2997   C_ wss 2999   (/)c0 3286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287

This theorem is referenced by:  undisj1  3340  undisj2  3341  disjpr2  3506