Theorem snelpw 4116

Description: A singleton of a set belongs to a power class of a set containing it. (Contributed by SF, 1-Feb-2015.)

Hypothesis

Ref	Expression
snelpw.1	|- A e. _V

Assertion

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

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. 2 |- A e. _V
2		snelpwg 4115	. 2 |- (A e. _V -> ({A} e. ~PB <-> A e. B))
3	1, 2	ax-mp 5	1 |- ({A} e. ~PB <-> A e. B)

Syntax hints:   <-> wb 176   e. wcel 1710  _Vcvv 2860  ~Pcpw 3723  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742

This theorem is referenced by:  sfinltfin  4536