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Theorem pw2eng 6984
Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal  2o. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
Assertion
Ref Expression
pw2eng  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )

Proof of Theorem pw2eng
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4210 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 ovex 5899 . . . 4  |-  ( {
(/) ,  { (/) } }  ^m  A )  e.  _V
32a1i 10 . . 3  |-  ( A  e.  V  ->  ( { (/) ,  { (/) } }  ^m  A )  e.  _V )
4 id 19 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
5 0ex 4166 . . . . 5  |-  (/)  e.  _V
65a1i 10 . . . 4  |-  ( A  e.  V  ->  (/)  e.  _V )
7 p0ex 4213 . . . . 5  |-  { (/) }  e.  _V
87a1i 10 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
9 0nep0 4197 . . . . 5  |-  (/)  =/=  { (/)
}
109a1i 10 . . . 4  |-  ( A  e.  V  ->  (/)  =/=  { (/)
} )
11 eqid 2296 . . . 4  |-  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) )  =  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/)
} ,  (/) ) ) )
124, 6, 8, 10, 11pw2f1o 6983 . . 3  |-  ( A  e.  V  ->  (
x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) ) : ~P A -1-1-onto-> ( {
(/) ,  { (/) } }  ^m  A ) )
13 f1oen2g 6894 . . 3  |-  ( ( ~P A  e.  _V  /\  ( { (/) ,  { (/)
} }  ^m  A
)  e.  _V  /\  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/)
} ,  (/) ) ) ) : ~P A -1-1-onto-> ( { (/) ,  { (/) } }  ^m  A ) )  ->  ~P A  ~~  ( { (/) ,  { (/)
} }  ^m  A
) )
141, 3, 12, 13syl3anc 1182 . 2  |-  ( A  e.  V  ->  ~P A  ~~  ( { (/) ,  { (/) } }  ^m  A ) )
15 df2o2 6509 . . 3  |-  2o  =  { (/) ,  { (/) } }
1615oveq1i 5884 . 2  |-  ( 2o 
^m  A )  =  ( { (/) ,  { (/)
} }  ^m  A
)
1714, 16syl6breqr 4079 1  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   {cpr 3654   class class class wbr 4039    e. cmpt 4093   -1-1-onto->wf1o 5270  (class class class)co 5874   2oc2o 6489    ^m cmap 6788    ~~ cen 6876
This theorem is referenced by:  pw2en  6985  pwen  7050  mappwen  7755  pwcdaen  7827  hauspwdom  17243
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-2o 6496  df-map 6790  df-en 6880
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