Theorem disjssun 2324

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 2310	. . . . . . 7 |- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))
2		ax-4 973	. . . . . . 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 735	. . . . 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 2171	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
8	6, 7	syl6rbbr 539	. . 3 |- (((A i^i B) = (/) /\ x e. A) -> (x e. (B u. C) <-> x e. C))
9	8	ralbidva 1658	. 2 |- ((A i^i B) = (/) -> (A.x e. A x e. (B u. C) <-> A.x e. A x e. C))
10		dfss3 2057	. 2 |- (A (_ (B u. C) <-> A.x e. A x e. (B u. C))
11		dfss3 2057	. 2 |- (A (_ C <-> A.x e. A x e. C)
12	9, 10, 11	3bitr4g 555	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 954   = wceq 956   e. wcel 958  A.wral 1644   u. cun 2043   i^i cin 2044   (_ wss 2045  (/)c0 2278

This theorem is referenced by:  ssxr 5528  cnfilca 10506

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-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279

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