Theorem pwelg 37865

Description: The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)

Assertion

Ref	Expression
pwelg	|- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> ( A e. B <-> ~P A e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem pwelg

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( U. x e. B /\ ~P x e. B ) -> ~P x e. B )
2	1	ralimi 2952	. . 3 |- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> A. x e. B ~P x e. B )
3		pweq 4161	. . . . 5 |- ( x = A -> ~P x = ~P A )
4	3	eleq1d 2686	. . . 4 |- ( x = A -> ( ~P x e. B <-> ~P A e. B ) )
5	4	rspccv 3306	. . 3 |- ( A. x e. B ~P x e. B -> ( A e. B -> ~P A e. B ) )
6	2, 5	syl 17	. 2 |- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> ( A e. B -> ~P A e. B ) )
7		simpl 473	. . . 4 |- ( ( U. x e. B /\ ~P x e. B ) -> U. x e. B )
8	7	ralimi 2952	. . 3 |- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> A. x e. B U. x e. B )
9		unieq 4444	. . . . . 6 |- ( x = ~P A -> U. x = U. ~P A )
10		unipw 4918	. . . . . 6 |- U. ~P A = A
11	9, 10	syl6eq 2672	. . . . 5 |- ( x = ~P A -> U. x = A )
12	11	eleq1d 2686	. . . 4 |- ( x = ~P A -> ( U. x e. B <-> A e. B ) )
13	12	rspccv 3306	. . 3 |- ( A. x e. B U. x e. B -> ( ~P A e. B -> A e. B ) )
14	8, 13	syl 17	. 2 |- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> ( ~P A e. B -> A e. B ) )
15	6, 14	impbid 202	1 |- ( A. x e. B ( U. x e. B /\ ~P x e. B ) -> ( A e. B <-> ~P A e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  pwinfig  37866