Theorem elpwpw 3899

Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
elpwpw	|- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )

Proof of Theorem elpwpw

Step	Hyp	Ref	Expression
1		elpwb 3520	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ A C_ ~P B ) )
2		sspwuni 3897	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	anbi2i 452	. 2 |- ( ( A e. _V /\ A C_ ~P B ) <-> ( A e. _V /\ U. A C_ B ) )
4	1, 3	bitri 183	1 |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   _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:  pwpwab  3900  elpwpwel  4396