Theorem elpwb 3662

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 2814	. 2 |- ( A e. ~P B -> A e. _V )
2		elpwg 3660	. 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 2202   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  elpwpw  4057  elpwpwel  4572