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Theorem pw2eng 7642
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 4640 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 ovex 6324 . . . 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 4587 . . . . 5  |-  (/)  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  e.  _V )
7 p0ex 4643 . . . . 5  |-  { (/) }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
9 0nep0 4627 . . . . 5  |-  (/)  =/=  { (/)
}
109a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  =/=  { (/)
} )
11 eqid 2457 . . . 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 7641 . . 3  |-  ( A  e.  V  ->  (
x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) ) : ~P A -1-1-onto-> ( {
(/) ,  { (/) } }  ^m  A ) )
13 f1oen2g 7551 . . 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 1228 . 2  |-  ( A  e.  V  ->  ~P A  ~~  ( { (/) ,  { (/) } }  ^m  A ) )
15 df2o2 7162 . . 3  |-  2o  =  { (/) ,  { (/) } }
1615oveq1i 6306 . 2  |-  ( 2o 
^m  A )  =  ( { (/) ,  { (/)
} }  ^m  A
)
1714, 16syl6breqr 4496 1  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793   ifcif 3944   ~Pcpw 4015   {csn 4032   {cpr 4034   class class class wbr 4456    |-> cmpt 4515   -1-1-onto->wf1o 5593  (class class class)co 6296   2oc2o 7142    ^m cmap 7438    ~~ cen 7532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1o 7148  df-2o 7149  df-map 7440  df-en 7536
This theorem is referenced by:  pw2en  7643  pwen  7709  mappwen  8510  pwcdaen  8582  hauspwdom  20127
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