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Theorem pwen 7272
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 7106 . . . 4  |-  Rel  ~~
21brrelexi 4910 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
3 pw2eng 7206 . . 3  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
42, 3syl 16 . 2  |-  ( A 
~~  B  ->  ~P A  ~~  ( 2o  ^m  A ) )
5 2onn 6875 . . . . . 6  |-  2o  e.  om
65elexi 2957 . . . . 5  |-  2o  e.  _V
76enref 7132 . . . 4  |-  2o  ~~  2o
8 mapen 7263 . . . 4  |-  ( ( 2o  ~~  2o  /\  A  ~~  B )  -> 
( 2o  ^m  A
)  ~~  ( 2o  ^m  B ) )
97, 8mpan 652 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ( 2o  ^m  B ) )
101brrelex2i 4911 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
11 pw2eng 7206 . . . 4  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
12 ensym 7148 . . . 4  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
1310, 11, 123syl 19 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  B )  ~~  ~P B )
14 entr 7151 . . 3  |-  ( ( ( 2o  ^m  A
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( 2o  ^m  A )  ~~  ~P B )
159, 13, 14syl2anc 643 . 2  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ~P B )
16 entr 7151 . 2  |-  ( ( ~P A  ~~  ( 2o  ^m  A )  /\  ( 2o  ^m  A ) 
~~  ~P B )  ->  ~P A  ~~  ~P B
)
174, 15, 16syl2anc 643 1  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204   omcom 4837  (class class class)co 6073   2oc2o 6710    ^m cmap 7010    ~~ cen 7098
This theorem is referenced by:  pwfi  7394  dfac12k  8019  pwcdaidm  8067  pwsdompw  8076  ackbij2lem2  8112  engch  8495  gchdomtri  8496  canthp1lem1  8519  gchcdaidm  8535  gchxpidm  8536  gchhar  8538  gchpwdom  8541  inar1  8642  rexpen  12819
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-3or 937  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-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-1st 6341  df-2nd 6342  df-1o 6716  df-2o 6717  df-er 6897  df-map 7012  df-en 7102
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