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Theorem pwen 7751
Description: If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
pwen  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )

Proof of Theorem pwen
StepHypRef Expression
1 relen 7582 . . . 4  |-  Rel  ~~
21brrelexi 4895 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
3 pw2eng 7684 . . 3  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
42, 3syl 17 . 2  |-  ( A 
~~  B  ->  ~P A  ~~  ( 2o  ^m  A ) )
5 2onn 7349 . . . . . 6  |-  2o  e.  om
65elexi 3097 . . . . 5  |-  2o  e.  _V
76enref 7609 . . . 4  |-  2o  ~~  2o
8 mapen 7742 . . . 4  |-  ( ( 2o  ~~  2o  /\  A  ~~  B )  -> 
( 2o  ^m  A
)  ~~  ( 2o  ^m  B ) )
97, 8mpan 674 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ( 2o  ^m  B ) )
101brrelex2i 4896 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
11 pw2eng 7684 . . . 4  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
12 ensym 7625 . . . 4  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
1310, 11, 123syl 18 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  B )  ~~  ~P B )
14 entr 7628 . . 3  |-  ( ( ( 2o  ^m  A
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( 2o  ^m  A )  ~~  ~P B )
159, 13, 14syl2anc 665 . 2  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ~P B )
16 entr 7628 . 2  |-  ( ( ~P A  ~~  ( 2o  ^m  A )  /\  ( 2o  ^m  A ) 
~~  ~P B )  ->  ~P A  ~~  ~P B
)
174, 15, 16syl2anc 665 1  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087   ~Pcpw 3985   class class class wbr 4426  (class class class)co 6305   omcom 6706   2oc2o 7184    ^m cmap 7480    ~~ cen 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-1o 7190  df-2o 7191  df-er 7371  df-map 7482  df-en 7578
This theorem is referenced by:  pwfi  7875  dfac12k  8575  pwcdaidm  8623  pwsdompw  8632  ackbij2lem2  8668  engch  9052  gchdomtri  9053  canthp1lem1  9076  gchcdaidm  9092  gchxpidm  9093  gchpwdom  9094  gchhar  9103  inar1  9199  rexpen  14258
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