Theorem snelpw 4298

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 3803	. 2 |- ( A e. B <-> { A } C_ B )
3	1	snex 4269	. . 3 |- { A } e. _V
4	3	elpw 3655	. 2 |- ( { A } e. ~P B <-> { A } C_ B )
5	2, 4	bitr4i 187	1 |- ( A e. B <-> { A } e. ~P B )

Syntax hints:   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by: (None)