Theorem elpwun 3855

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. ~PB)

Proof of Theorem elpwun

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. ~P(B u. C) -> A e. _V)
2		elisset 2299	. . 3 |- ((A \ C) e. ~PB -> (A \ C) e. _V)
3		eldifpw.1	. . . 4 |- C e. _V
4		difex2 3802	. . . 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 217	. 2 |- ((A \ C) e. ~PB -> A e. _V)
7		elpwg 3038	. . 3 |- (A e. _V -> (A e. ~P(B u. C) <-> A C_ (B u. C)))
8		difexg 3458	. . . . 5 |- (A e. _V -> (A \ C) e. _V)
9		elpwg 3038	. . . . 5 |- ((A \ C) e. _V -> ((A \ C) e. ~PB <-> (A \ C) C_ B))
10	8, 9	syl 12	. . . 4 |- (A e. _V -> ((A \ C) e. ~PB <-> (A \ C) C_ B))
11		uncom 2744	. . . . . 6 |- (B u. C) = (C u. B)
12	11	sseq2i 2642	. . . . 5 |- (A C_ (B u. C) <-> A C_ (C u. B))
13		ssundif 2955	. . . . 5 |- (A C_ (C u. B) <-> (A \ C) C_ B)
14	12, 13	bitri 190	. . . 4 |- (A C_ (B u. C) <-> (A \ C) C_ B)
15	10, 14	syl6rbbr 598	. . 3 |- (A e. _V -> (A C_ (B u. C) <-> (A \ C) e. ~PB))
16	7, 15	bitrd 587	. 2 |- (A e. _V -> (A e. ~P(B u. C) <-> (A \ C) e. ~PB))
17	1, 6, 16	pm5.21nii 743	1 |- (A e. ~P(B u. C) <-> (A \ C) e. ~PB)

Syntax hints:   <-> wb 163   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  ~Pcpw 3032

This theorem is referenced by:  pwfilem 5660

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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