Theorem elpwb 3612

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 2771	. 2 |- ( A e. ~P B -> A e. _V )
2		elpwg 3610	. 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 2164   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-pw 3604

This theorem is referenced by:  elpwpw  4000  elpwpwel  4507