Theorem un00 3515

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 3330	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3502	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	eqtrdi 2256	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3344	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3225	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3508	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 122	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3345	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3225	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3508	. . . 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 3172   C_ wss 3174   (/)c0 3468

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  undisj1  3526  undisj2  3527  disjpr2  3707