Theorem pwundif 3579

Description: Break up the power class of a union into a union of smaller classes.

Assertion

Ref	Expression
pwundif	|- ~P(A u. B) = ((~P(A u. B) \ ~PA) u. ~PA)

Proof of Theorem pwundif

Step	Hyp	Ref	Expression
1		visset 2295	. . . 4 |- x e. _V
2	1	elpw 3037	. . 3 |- (x e. ~P(A u. B) <-> x C_ (A u. B))
3		elun 2741	. . . 4 |- (x e. ((~P(A u. B) \ ~PA) u. ~PA) <-> (x e. (~P(A u. B) \ ~PA) \/ x e. ~PA))
4		eldif 2609	. . . . . 6 |- (x e. (~P(A u. B) \ ~PA) <-> (x e. ~P(A u. B) /\ -. x e. ~PA))
5	1	elpw 3037	. . . . . . . 8 |- (x e. ~PA <-> x C_ A)
6	5	notbii 204	. . . . . . 7 |- (-. x e. ~PA <-> -. x C_ A)
7	2, 6	anbi12i 540	. . . . . 6 |- ((x e. ~P(A u. B) /\ -. x e. ~PA) <-> (x C_ (A u. B) /\ -. x C_ A))
8	4, 7	bitri 190	. . . . 5 |- (x e. (~P(A u. B) \ ~PA) <-> (x C_ (A u. B) /\ -. x C_ A))
9	8, 5	orbi12i 277	. . . 4 |- ((x e. (~P(A u. B) \ ~PA) \/ x e. ~PA) <-> ((x C_ (A u. B) /\ -. x C_ A) \/ x C_ A))
10		ordir 658	. . . . 5 |- (((x C_ (A u. B) /\ -. x C_ A) \/ x C_ A) <-> ((x C_ (A u. B) \/ x C_ A) /\ (-. x C_ A \/ x C_ A)))
11		pm2.1 718	. . . . . 6 |- (-. x C_ A \/ x C_ A)
12	11	biantru 793	. . . . 5 |- ((x C_ (A u. B) \/ x C_ A) <-> ((x C_ (A u. B) \/ x C_ A) /\ (-. x C_ A \/ x C_ A)))
13		id 73	. . . . . . 7 |- (x C_ (A u. B) -> x C_ (A u. B))
14		ssun3 2769	. . . . . . 7 |- (x C_ A -> x C_ (A u. B))
15	13, 14	jaoi 368	. . . . . 6 |- ((x C_ (A u. B) \/ x C_ A) -> x C_ (A u. B))
16		orc 291	. . . . . 6 |- (x C_ (A u. B) -> (x C_ (A u. B) \/ x C_ A))
17	15, 16	impbii 174	. . . . 5 |- ((x C_ (A u. B) \/ x C_ A) <-> x C_ (A u. B))
18	10, 12, 17	3bitr2i 196	. . . 4 |- (((x C_ (A u. B) /\ -. x C_ A) \/ x C_ A) <-> x C_ (A u. B))
19	3, 9, 18	3bitrri 195	. . 3 |- (x C_ (A u. B) <-> x e. ((~P(A u. B) \ ~PA) u. ~PA))
20	2, 19	bitri 190	. 2 |- (x e. ~P(A u. B) <-> x e. ((~P(A u. B) \ ~PA) u. ~PA))
21	20	eqriv 1881	1 |- ~P(A u. B) = ((~P(A u. B) \ ~PA) u. ~PA)

Syntax hints:  -. wn 2   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   \ 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-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

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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pw 3035

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