Theorem un00 1728

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 1613	. . 3 |- ((A = (/) /\ B = (/)) -> (A u. B) = ((/) u. (/)))
2		un0 1721	. . 3 |- ((/) u. (/)) = (/)
3	1, 2	syl6eq 1140	. 2 |- ((A = (/) /\ B = (/)) -> (A u. B) = (/))
4		ssun1 1621	. . . . 5 |- A (_ (A u. B)
5		sseq2 1522	. . . . 5 |- ((A u. B) = (/) -> (A (_ (A u. B) <-> A (_ (/)))
6	4, 5	mpbii 168	. . . 4 |- ((A u. B) = (/) -> A (_ (/))
7		ss0b 1726	. . . 4 |- (A (_ (/) <-> A = (/))
8	6, 7	sylib 173	. . 3 |- ((A u. B) = (/) -> A = (/))
9		ssun2 1622	. . . . 5 |- B (_ (A u. B)
10		sseq2 1522	. . . . 5 |- ((A u. B) = (/) -> (B (_ (A u. B) <-> B (_ (/)))
11	9, 10	mpbii 168	. . . 4 |- ((A u. B) = (/) -> B (_ (/))
12		ss0b 1726	. . . 4 |- (B (_ (/) <-> B = (/))
13	11, 12	sylib 173	. . 3 |- ((A u. B) = (/) -> B = (/))
14	8, 13	jca 236	. 2 |- ((A u. B) = (/) -> (A = (/) /\ B = (/)))
15	3, 14	impbi 139	1 |- ((A = (/) /\ B = (/)) <-> (A u. B) = (/))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   u. cun 1485   (_ wss 1487  (/)c0 1707

This theorem is referenced by:  undisj1 1739  undisj2 1740

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

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