Theorem elpwb 3636

Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)

Assertion

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

Proof of Theorem elpwb

Step	Hyp	Ref	Expression
1		elex 2788	. 2 |- ( A e. ~P B -> A e. _V )
2		elpwg 3634	. 2 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	biadan2 456	1 |- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2178   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  elpwpw  4028  elpwpwel  4540