Theorem un00 2296

Description: Two classes are empty iff their union is empty.

Assertion

Ref	Expression
un00	|- ((A = (/) /\ B = (/)) <-> (A u. B) = (/))

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 2169	. . 3 |- ((A = (/) /\ B = (/)) -> (A u. B) = ((/) u. (/)))
2		un0 2287	. . 3 |- ((/) u. (/)) = (/)
3	1, 2	syl6eq 1515	. 2 |- ((A = (/) /\ B = (/)) -> (A u. B) = (/))
4		ssun1 2183	. . . . 5 |- A (_ (A u. B)
5		sseq2 2073	. . . . 5 |- ((A u. B) = (/) -> (A (_ (A u. B) <-> A (_ (/)))
6	4, 5	mpbii 193	. . . 4 |- ((A u. B) = (/) -> A (_ (/))
7		ss0b 2292	. . . 4 |- (A (_ (/) <-> A = (/))
8	6, 7	sylib 198	. . 3 |- ((A u. B) = (/) -> A = (/))
9		ssun2 2184	. . . . 5 |- B (_ (A u. B)
10		sseq2 2073	. . . . 5 |- ((A u. B) = (/) -> (B (_ (A u. B) <-> B (_ (/)))
11	9, 10	mpbii 193	. . . 4 |- ((A u. B) = (/) -> B (_ (/))
12		ss0b 2292	. . . 4 |- (B (_ (/) <-> B = (/))
13	11, 12	sylib 198	. . 3 |- ((A u. B) = (/) -> B = (/))
14	8, 13	jca 288	. 2 |- ((A u. B) = (/) -> (A = (/) /\ B = (/)))
15	3, 14	impbi 157	1 |- ((A = (/) /\ B = (/)) <-> (A u. B) = (/))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   u. cun 2035   (_ wss 2037  (/)c0 2270

This theorem is referenced by:  undisj1 2310  undisj2 2311  rankxplim3 4686  cnfilca 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271

Copyright terms: Public domain