Theorem elpwpw 4094

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 3695	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ A C_ ~P B ) )
2		sspwuni 4092	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	anbi2i 461	. 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 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931

This theorem is referenced by:  pwpwab  4095  elpwpwel  4616