Theorem snelpw 2752

Description: A singleton of a set belongs to the power class of a class containing the set.

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 2461	. 2 |- (A e. B <-> {A} (_ B)
3		snex 2750	. . 3 |- {A} e. V
4	3	elpw 2404	. 2 |- ({A} e. P~B <-> {A} (_ B)
5	2, 4	bitr4 176	1 |- (A e. B <-> {A} e. P~B)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401  {csn 2409

This theorem is referenced by:  unipw 2756  canth2 4484  abfi 10451  abfiOLD 10452  dtt2 10618

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412

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