Theorem elpwb 3695

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 2833	. 2 |- ( A e. ~P B -> A e. _V )
2		elpwg 3693	. 2 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	biadan2 460	1 |- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685

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-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  elpwpw  4094  elpwpwel  4616