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Theorem pw2eng 7206
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 4375 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 ovex 6098 . . . 4  |-  ( {
(/) ,  { (/) } }  ^m  A )  e.  _V
32a1i 11 . . 3  |-  ( A  e.  V  ->  ( { (/) ,  { (/) } }  ^m  A )  e.  _V )
4 id 20 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
5 0ex 4331 . . . . 5  |-  (/)  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  e.  _V )
7 p0ex 4378 . . . . 5  |-  { (/) }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
9 0nep0 4362 . . . . 5  |-  (/)  =/=  { (/)
}
109a1i 11 . . . 4  |-  ( A  e.  V  ->  (/)  =/=  { (/)
} )
11 eqid 2435 . . . 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 7205 . . 3  |-  ( A  e.  V  ->  (
x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  { (/) } ,  (/) ) ) ) : ~P A -1-1-onto-> ( {
(/) ,  { (/) } }  ^m  A ) )
13 f1oen2g 7116 . . 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 1184 . 2  |-  ( A  e.  V  ->  ~P A  ~~  ( { (/) ,  { (/) } }  ^m  A ) )
15 df2o2 6730 . . 3  |-  2o  =  { (/) ,  { (/) } }
1615oveq1i 6083 . 2  |-  ( 2o 
^m  A )  =  ( { (/) ,  { (/)
} }  ^m  A
)
1714, 16syl6breqr 4244 1  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   ifcif 3731   ~Pcpw 3791   {csn 3806   {cpr 3807   class class class wbr 4204    e. cmpt 4258   -1-1-onto->wf1o 5445  (class class class)co 6073   2oc2o 6710    ^m cmap 7010    ~~ cen 7098
This theorem is referenced by:  pw2en  7207  pwen  7272  mappwen  7985  pwcdaen  8057  hauspwdom  17556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1o 6716  df-2o 6717  df-map 7012  df-en 7102
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