Theorem un00 3587

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 3414	. . 3 |- ((A = (/) /\ B = (/)) -> (A u. B) = ((/) u. (/)))
2		un0 3576	. . 3 |- ((/) u. (/)) = (/)
3	1, 2	syl6eq 2401	. 2 |- ((A = (/) /\ B = (/)) -> (A u. B) = (/))
4		ssun1 3427	. . . . 5 |- A C_ (A u. B)
5		sseq2 3294	. . . . 5 |- ((A u. B) = (/) -> (A C_ (A u. B) <-> A C_ (/)))
6	4, 5	mpbii 202	. . . 4 |- ((A u. B) = (/) -> A C_ (/))
7		ss0b 3581	. . . 4 |- (A C_ (/) <-> A = (/))
8	6, 7	sylib 188	. . 3 |- ((A u. B) = (/) -> A = (/))
9		ssun2 3428	. . . . 5 |- B C_ (A u. B)
10		sseq2 3294	. . . . 5 |- ((A u. B) = (/) -> (B C_ (A u. B) <-> B C_ (/)))
11	9, 10	mpbii 202	. . . 4 |- ((A u. B) = (/) -> B C_ (/))
12		ss0b 3581	. . . 4 |- (B C_ (/) <-> B = (/))
13	11, 12	sylib 188	. . 3 |- ((A u. B) = (/) -> B = (/))
14	8, 13	jca 518	. 2 |- ((A u. B) = (/) -> (A = (/) /\ B = (/)))
15	3, 14	impbii 180	1 |- ((A = (/) /\ B = (/)) <-> (A u. B) = (/))

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   u. cun 3208   C_ wss 3258  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  undisj1  3603  undisj2  3604  0nelsuc  4401  addcass  4416