Theorem pwpwab 3900

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 2689	. . 3 |- x e. _V
2		elpwpw 3899	. . 3 |- ( x e. ~P ~P A <-> ( x e. _V /\ U. x C_ A ) )
3	1, 2	mpbiran 924	. 2 |- ( x e. ~P ~P A <-> U. x C_ A )
4	3	abbi2i 2254	1 |- ~P ~P A = { x | U. x C_ A }

Syntax hints:   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   C_ wss 3071   ~Pcpw 3510   U.cuni 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737

This theorem is referenced by:  pwpwssunieq  3901