Theorem elpwun 2901

Description: Membership in the power class of a union.

Hypothesis

Ref	Expression
eldifpw.1	|- C e. V

Assertion

Ref	Expression
elpwun	|- (A e. P~(B u. C) <-> (A \ C) e. P~B)

Proof of Theorem elpwun

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. P~(B u. C) -> A e. V)
2		elisset 1808	. . 3 |- ((A \ C) e. P~B -> (A \ C) e. V)
3		eldifpw.1	. . . 4 |- C e. V
4		difex2 2867	. . . 4 |- (C e. V -> (A e. V <-> (A \ C) e. V))
5	3, 4	ax-mp 7	. . 3 |- (A e. V <-> (A \ C) e. V)
6	2, 5	sylibr 200	. 2 |- ((A \ C) e. P~B -> A e. V)
7		elpwg 2395	. . 3 |- (A e. V -> (A e. P~(B u. C) <-> A (_ (B u. C)))
8		difexg 2712	. . . . 5 |- (A e. V -> (A \ C) e. V)
9		elpwg 2395	. . . . 5 |- ((A \ C) e. V -> ((A \ C) e. P~B <-> (A \ C) (_ B))
10	8, 9	syl 10	. . . 4 |- (A e. V -> ((A \ C) e. P~B <-> (A \ C) (_ B))
11		uncom 2166	. . . . . 6 |- (B u. C) = (C u. B)
12	11	sseq2i 2076	. . . . 5 |- (A (_ (B u. C) <-> A (_ (C u. B))
13		ssundif 2334	. . . . 5 |- (A (_ (C u. B) <-> (A \ C) (_ B)
14	12, 13	bitr 173	. . . 4 |- (A (_ (B u. C) <-> (A \ C) (_ B)
15	10, 14	syl6rbbr 537	. . 3 |- (A e. V -> (A (_ (B u. C) <-> (A \ C) e. P~B))
16	7, 15	bitrd 526	. 2 |- (A e. V -> (A e. P~(B u. C) <-> (A \ C) e. P~B))
17	1, 6, 16	pm5.21nii 677	1 |- (A e. P~(B u. C) <-> (A \ C) e. P~B)

Syntax hints:   <-> wb 146   e. wcel 955  Vcvv 1802   \ cdif 2034   u. cun 2035   (_ wss 2037  P~cpw 2391

This theorem is referenced by:  pwfilem 4544

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494

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