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Theorem pw2eng 7696
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 4585 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 ovex 6336 . . . 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 4528 . . . . 5  |-  (/)  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  e.  _V )
7 p0ex 4588 . . . . 5  |-  { (/) }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
9 0nep0 4572 . . . . 5  |-  (/)  =/=  { (/)
}
109a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  =/=  { (/)
} )
11 eqid 2471 . . . 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 7695 . . 3  |-  ( A  e.  V  ->  (
x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) ) : ~P A -1-1-onto-> ( {
(/) ,  { (/) } }  ^m  A ) )
13 f1oen2g 7604 . . 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 1292 . 2  |-  ( A  e.  V  ->  ~P A  ~~  ( { (/) ,  { (/) } }  ^m  A ) )
15 df2o2 7214 . . 3  |-  2o  =  { (/) ,  { (/) } }
1615oveq1i 6318 . 2  |-  ( 2o 
^m  A )  =  ( { (/) ,  { (/)
} }  ^m  A
)
1714, 16syl6breqr 4436 1  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   -1-1-onto->wf1o 5588  (class class class)co 6308   2oc2o 7194    ^m cmap 7490    ~~ cen 7584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1o 7200  df-2o 7201  df-map 7492  df-en 7588
This theorem is referenced by:  pw2en  7697  pwen  7763  mappwen  8561  pwcdaen  8633  hauspwdom  20593
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