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Theorem pw2eng 7417
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 4476 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 ovex 6116 . . . 4  |-  ( {
(/) ,  { (/) } }  ^m  A )  e.  _V
32a1i 11 . . 3  |-  ( A  e.  V  ->  ( { (/) ,  { (/) } }  ^m  A )  e.  _V )
4 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
5 0ex 4422 . . . . 5  |-  (/)  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  e.  _V )
7 p0ex 4479 . . . . 5  |-  { (/) }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
9 0nep0 4463 . . . . 5  |-  (/)  =/=  { (/)
}
109a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  =/=  { (/)
} )
11 eqid 2443 . . . 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 7416 . . 3  |-  ( A  e.  V  ->  (
x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) ) : ~P A -1-1-onto-> ( {
(/) ,  { (/) } }  ^m  A ) )
13 f1oen2g 7326 . . 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 1218 . 2  |-  ( A  e.  V  ->  ~P A  ~~  ( { (/) ,  { (/) } }  ^m  A ) )
15 df2o2 6934 . . 3  |-  2o  =  { (/) ,  { (/) } }
1615oveq1i 6101 . 2  |-  ( 2o 
^m  A )  =  ( { (/) ,  { (/)
} }  ^m  A
)
1714, 16syl6breqr 4332 1  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    =/= wne 2606   _Vcvv 2972   (/)c0 3637   ifcif 3791   ~Pcpw 3860   {csn 3877   {cpr 3879   class class class wbr 4292    e. cmpt 4350   -1-1-onto->wf1o 5417  (class class class)co 6091   2oc2o 6914    ^m cmap 7214    ~~ cen 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1o 6920  df-2o 6921  df-map 7216  df-en 7311
This theorem is referenced by:  pw2en  7418  pwen  7484  mappwen  8282  pwcdaen  8354  hauspwdom  19105
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