Theorem elpwb 3576

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 2741	. 2 |- ( A e. ~P B -> A e. _V )
2		elpwg 3574	. 2 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	biadan2 453	1 |- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  elpwpw  3959  elpwpwel  4460