Theorem snelpw 4185

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 3696	. 2 |- ( A e. B <-> { A } C_ B )
3	1	snex 4158	. . 3 |- { A } e. _V
4	3	elpw 3559	. 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 2135   _Vcvv 2721   C_ wss 3111   ~Pcpw 3553   {csn 3570

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576

This theorem is referenced by: (None)