Theorem elpwpw 3952

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 3569	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ A C_ ~P B ) )
2		sspwuni 3950	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	anbi2i 453	. 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 2136   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   U.cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-uni 3790

This theorem is referenced by:  pwpwab  3953  elpwpwel  4453