Theorem pwpwuni 39225

Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwpwuni	|- ( A e. V -> ( A e. ~P ~P B <-> U. A e. ~P B ) )

Proof of Theorem pwpwuni

Step	Hyp	Ref	Expression
1		elpwg 4166	. 2 |- ( A e. V -> ( A e. ~P ~P B <-> A C_ ~P B ) )
2		sspwuni 4611	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	a1i 11	. 2 |- ( A e. V -> ( A C_ ~P B <-> U. A C_ B ) )
4		uniexg 6955	. . . 4 |- ( A e. V -> U. A e. _V )
5		elpwg 4166	. . . 4 |- ( U. A e. _V -> ( U. A e. ~P B <-> U. A C_ B ) )
6	4, 5	syl 17	. . 3 |- ( A e. V -> ( U. A e. ~P B <-> U. A C_ B ) )
7	6	bicomd 213	. 2 |- ( A e. V -> ( U. A C_ B <-> U. A e. ~P B ) )
8	1, 3, 7	3bitrd 294	1 |- ( A e. V -> ( A e. ~P ~P B <-> U. A e. ~P B ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  psmeasurelem  40687