Theorem un00 3471

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 3286	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3458	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	eqtrdi 2226	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3300	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3181	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3464	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 122	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3301	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3181	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 148	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3464	. . . 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 1353   u. cun 3129   C_ wss 3131   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425

This theorem is referenced by:  undisj1  3482  undisj2  3483  disjpr2  3658