Theorem ssundif 2344

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		pm5.6 688	. . . 4 |- (((x e. A /\ -. x e. B) -> x e. C) <-> (x e. A -> (x e. B \/ x e. C)))
2		eldif 2057	. . . . 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 2173	. . . . 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 999	. 2 |- (A.x(x e. A -> x e. (B u. C)) <-> A.x(x e. (A \ B) -> x e. C))
8		dfss2 2058	. 2 |- (A (_ (B u. C) <-> A.x(x e. A -> x e. (B u. C)))
9		dfss2 2058	. 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 954   e. wcel 958   \ cdif 2044   u. cun 2045   (_ wss 2047

This theorem is referenced by:  difcom 2345  elpwun 2911  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

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