Theorem elpwun 2917

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 1820	. 2 |- (A e. P~(B u. C) -> A e. V)
2		elisset 1820	. . 3 |- ((A \ C) e. P~B -> (A \ C) e. V)
3		eldifpw.1	. . . 4 |- C e. V
4		difex2 2883	. . . 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 2409	. . 3 |- (A e. V -> (A e. P~(B u. C) <-> A (_ (B u. C)))
8		difexg 2727	. . . . 5 |- (A e. V -> (A \ C) e. V)
9		elpwg 2409	. . . . 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 2179	. . . . . 6 |- (B u. C) = (C u. B)
12	11	sseq2i 2089	. . . . 5 |- (A (_ (B u. C) <-> A (_ (C u. B))
13		ssundif 2348	. . . . 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 541	. . 3 |- (A e. V -> (A (_ (B u. C) <-> (A \ C) e. P~B))
16	7, 15	bitrd 530	. 2 |- (A e. V -> (A e. P~(B u. C) <-> (A \ C) e. P~B))
17	1, 6, 16	pm5.21nii 681	1 |- (A e. P~(B u. C) <-> (A \ C) e. P~B)

Syntax hints:   <-> wb 146   e. wcel 960  Vcvv 1814   \ cdif 2047   u. cun 2048   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  pwfilem 4577  pwfilemOLD 4578

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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