Theorem elpwpw 4613

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 4169	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ A C_ ~P B ) )
2		sspwuni 4611	. . 3 |- ( A C_ ~P B <-> U. A C_ B )
3	2	anbi2i 730	. 2 |- ( ( A e. _V /\ A C_ ~P B ) <-> ( A e. _V /\ U. A C_ B ) )
4	1, 3	bitri 264	1 |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  pwpwab  4614