Theorem pwpwab 4614

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	|- ~P ~P A = { x | U. x C_ A }

Distinct variable group:   x, A

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 3203	. . 3 |- x e. _V
2		elpwpw 4613	. . 3 |- ( x e. ~P ~P A <-> ( x e. _V /\ U. x C_ A ) )
3	1, 2	mpbiran 953	. 2 |- ( x e. ~P ~P A <-> U. x C_ A )
4	3	abbi2i 2738	1 |- ~P ~P A = { x | U. x C_ A }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   _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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  pwpwssunieq  4615