Theorem snelpw 4306

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 3809	. 2 |- ( A e. B <-> { A } C_ B )
3	1	snex 4277	. . 3 |- { A } e. _V
4	3	elpw 3659	. 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 2201   _Vcvv 2801   C_ wss 3199   ~Pcpw 3653   {csn 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676

This theorem is referenced by: (None)