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Theorem snelpwg 25091
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
snelpwg  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  e.  ~P B
) )

Proof of Theorem snelpwg
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 eleq1 2343 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( A  e.  B  <->  if ( A  e.  _V ,  A ,  (/) )  e.  B
) )
3 sneq 3651 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  { A }  =  { if ( A  e.  _V ,  A ,  (/) ) } )
43eleq1d 2349 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( { A }  e.  ~P B 
<->  { if ( A  e.  _V ,  A ,  (/) ) }  e.  ~P B ) )
52, 4bibi12d 312 . . 3  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( A  e.  B  <->  { A }  e.  ~P B )  <->  ( if ( A  e.  _V ,  A ,  (/) )  e.  B  <->  { if ( A  e.  _V ,  A ,  (/) ) }  e.  ~P B ) ) )
6 0ex 4150 . . . . 5  |-  (/)  e.  _V
76elimel 3617 . . . 4  |-  if ( A  e.  _V ,  A ,  (/) )  e. 
_V
87snelpw 4221 . . 3  |-  ( if ( A  e.  _V ,  A ,  (/) )  e.  B  <->  { if ( A  e.  _V ,  A ,  (/) ) }  e.  ~P B )
95, 8dedth 3606 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  { A }  e.  ~P B
) )
101, 9syl 15 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  e.  ~P B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647
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