Theorem snelpw 4073

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Hypothesis

Ref	Expression
snelpw.1	|- A e. _V

Assertion

Ref	Expression
snelpw	|- ( A e. B <-> { A } e. ~P B )

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 |- A e. _V
2	1	snss 3596	. 2 |- ( A e. B <-> { A } C_ B )
3	1	snex 4049	. . 3 |- { A } e. _V
4	3	elpw 3463	. 2 |- ( { A } e. ~P B <-> { A } C_ B )
5	2, 4	bitr4i 186	1 |- ( A e. B <-> { A } e. ~P B )

Syntax hints:   <-> wb 104   e. wcel 1448   _Vcvv 2641   C_ wss 3021   ~Pcpw 3457   {csn 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480

This theorem is referenced by: (None)