Theorem ssundif 2340

Description: A condition equivalent to inclusion in the union of two classes.

Assertion

Ref	Expression
ssundif	|- (A (_ (B u. C) <-> (A \ B) (_ C)

Proof of Theorem ssundif

Step	Hyp	Ref	Expression
1		pm5.6 687	. . . 4 |- (((x e. A /\ -. x e. B) -> x e. C) <-> (x e. A -> (x e. B \/ x e. C)))
2		eldif 2053	. . . . 5 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
3	2	imbi1i 186	. . . 4 |- ((x e. (A \ B) -> x e. C) <-> ((x e. A /\ -. x e. B) -> x e. C))
4		elun 2169	. . . . 5 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	4	imbi2i 185	. . . 4 |- ((x e. A -> x e. (B u. C)) <-> (x e. A -> (x e. B \/ x e. C)))
6	1, 3, 5	3bitr4r 184	. . 3 |- ((x e. A -> x e. (B u. C)) <-> (x e. (A \ B) -> x e. C))
7	6	albii 997	. 2 |- (A.x(x e. A -> x e. (B u. C)) <-> A.x(x e. (A \ B) -> x e. C))
8		dfss2 2054	. 2 |- (A (_ (B u. C) <-> A.x(x e. A -> x e. (B u. C)))
9		dfss2 2054	. 2 |- ((A \ B) (_ C <-> A.x(x e. (A \ B) -> x e. C))
10	7, 8, 9	3bitr4 183	1 |- (A (_ (B u. C) <-> (A \ B) (_ C)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 952   e. wcel 956   \ cdif 2040   u. cun 2041   (_ wss 2043

This theorem is referenced by:  difcom 2341  elpwun 2906  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-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049

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