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Theorem snelpw 2742
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
StepHypRef Expression
1 snelpw.1 . . 3 |- A e. V
21snss 2452 . 2 |- (A e. B <-> {A} (_ B)
3 snex 2740 . . 3 |- {A} e. V
43elpw 2394 . 2 |- ({A} e. P~B <-> {A} (_ B)
52, 4bitr4 176 1 |- (A e. B <-> {A} e. P~B)
Colors of variables: wff set class
Syntax hints:   <-> wb 146   e. wcel 955  Vcvv 1802   (_ wss 2037  P~cpw 2391  {csn 2399
This theorem is referenced by:  unipw 2746  canth2 4464  abfi 10349  dtt2 10462
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402
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