Theorem un00 2306

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 2179	. . 3 |- ((A = (/) /\ B = (/)) -> (A u. B) = ((/) u. (/)))
2		un0 2297	. . 3 |- ((/) u. (/)) = (/)
3	1, 2	syl6eq 1523	. 2 |- ((A = (/) /\ B = (/)) -> (A u. B) = (/))
4		ssun1 2193	. . . . 5 |- A (_ (A u. B)
5		sseq2 2083	. . . . 5 |- ((A u. B) = (/) -> (A (_ (A u. B) <-> A (_ (/)))
6	4, 5	mpbii 193	. . . 4 |- ((A u. B) = (/) -> A (_ (/))
7		ss0b 2302	. . . 4 |- (A (_ (/) <-> A = (/))
8	6, 7	sylib 198	. . 3 |- ((A u. B) = (/) -> A = (/))
9		ssun2 2194	. . . . 5 |- B (_ (A u. B)
10		sseq2 2083	. . . . 5 |- ((A u. B) = (/) -> (B (_ (A u. B) <-> B (_ (/)))
11	9, 10	mpbii 193	. . . 4 |- ((A u. B) = (/) -> B (_ (/))
12		ss0b 2302	. . . 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 956   u. cun 2045   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  undisj1 2320  undisj2 2321  rankxplim3 4714  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281

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