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Theorem pwen 7002
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 6836 . . . 4  |-  Rel  ~~
21brrelexi 4717 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
3 pw2eng 6936 . . 3  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
42, 3syl 17 . 2  |-  ( A 
~~  B  ->  ~P A  ~~  ( 2o  ^m  A ) )
5 2onn 6606 . . . . . 6  |-  2o  e.  om
65elexi 2772 . . . . 5  |-  2o  e.  _V
76enref 6862 . . . 4  |-  2o  ~~  2o
8 mapen 6993 . . . 4  |-  ( ( 2o  ~~  2o  /\  A  ~~  B )  -> 
( 2o  ^m  A
)  ~~  ( 2o  ^m  B ) )
97, 8mpan 654 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ( 2o  ^m  B ) )
101brrelex2i 4718 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
11 pw2eng 6936 . . . 4  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
12 ensym 6878 . . . 4  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
1310, 11, 123syl 20 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  B )  ~~  ~P B )
14 entr 6881 . . 3  |-  ( ( ( 2o  ^m  A
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( 2o  ^m  A )  ~~  ~P B )
159, 13, 14syl2anc 645 . 2  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ~P B )
16 entr 6881 . 2  |-  ( ( ~P A  ~~  ( 2o  ^m  A )  /\  ( 2o  ^m  A ) 
~~  ~P B )  ->  ~P A  ~~  ~P B
)
174, 15, 16syl2anc 645 1  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2763   ~Pcpw 3599   class class class wbr 3997   omcom 4628  (class class class)co 5792   2oc2o 6441    ^m cmap 6740    ~~ cen 6828
This theorem is referenced by:  pwfi  7119  dfac12k  7741  pwcdaidm  7789  pwsdompw  7798  ackbij2lem2  7834  engch  8218  gchdomtri  8219  canthp1lem1  8242  gchcdaidm  8258  gchxpidm  8259  gchhar  8261  gchpwdom  8264  inar1  8365  rexpen  12468
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-1o 6447  df-2o 6448  df-er 6628  df-map 6742  df-en 6832
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