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Theorem snelpwg 25194
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 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 eleq1 2356 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( A  e.  B  <->  if ( A  e.  _V ,  A ,  (/) )  e.  B
) )
3 sneq 3664 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  { A }  =  { if ( A  e.  _V ,  A ,  (/) ) } )
43eleq1d 2362 . . . 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 4166 . . . . 5  |-  (/)  e.  _V
76elimel 3630 . . . 4  |-  if ( A  e.  _V ,  A ,  (/) )  e. 
_V
87snelpw 4237 . . 3  |-  ( if ( A  e.  _V ,  A ,  (/) )  e.  B  <->  { if ( A  e.  _V ,  A ,  (/) ) }  e.  ~P B )
95, 8dedth 3619 . 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660
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