Theorem elpwpw 4014

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 3626	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ A C_ ~P B ) )
2		sspwuni 4012	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	anbi2i 457	. 2 |- ( ( A e. _V /\ A C_ ~P B ) <-> ( A e. _V /\ U. A C_ B ) )
4	1, 3	bitri 184	1 |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2176   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   U.cuni 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-uni 3851

This theorem is referenced by:  pwpwab  4015  elpwpwel  4522