Theorem un00 3507

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 3322	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3494	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	eqtrdi 2254	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3336	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3217	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3500	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 122	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3337	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3217	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3500	. . . 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 1373   u. cun 3164   C_ wss 3166   (/)c0 3460

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  undisj1  3518  undisj2  3519  disjpr2  3697