Theorem disjssun 2322

Description: Subset relation for disjoint classes.

Assertion

Ref	Expression
disjssun	|- ((A i^i B) = (/) -> (A (_ (B u. C) <-> A (_ C))

Proof of Theorem disjssun

Step	Hyp	Ref	Expression
1		disj1 2308	. . . . . . 7 |- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))
2		ax-4 971	. . . . . . 7 |- (A.x(x e. A -> -. x e. B) -> (x e. A -> -. x e. B))
3	1, 2	sylbi 199	. . . . . 6 |- ((A i^i B) = (/) -> (x e. A -> -. x e. B))
4	3	imp 350	. . . . 5 |- (((A i^i B) = (/) /\ x e. A) -> -. x e. B)
5		biorf 734	. . . . 5 |- (-. x e. B -> (x e. C <-> (x e. B \/ x e. C)))
6	4, 5	syl 10	. . . 4 |- (((A i^i B) = (/) /\ x e. A) -> (x e. C <-> (x e. B \/ x e. C)))
7		elun 2169	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
8	6, 7	syl6rbbr 538	. . 3 |- (((A i^i B) = (/) /\ x e. A) -> (x e. (B u. C) <-> x e. C))
9	8	ralbidva 1656	. 2 |- ((A i^i B) = (/) -> (A.x e. A x e. (B u. C) <-> A.x e. A x e. C))
10		dfss3 2055	. 2 |- (A (_ (B u. C) <-> A.x e. A x e. (B u. C))
11		dfss3 2055	. 2 |- (A (_ C <-> A.x e. A x e. C)
12	9, 10, 11	3bitr4g 554	1 |- ((A i^i B) = (/) -> (A (_ (B u. C) <-> A (_ C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642   u. cun 2041   i^i cin 2042   (_ wss 2043  (/)c0 2276

This theorem is referenced by:  ssxr 5521  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277

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