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Theorem pwen 7243
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 7077 . . . 4  |-  Rel  ~~
21brrelexi 4881 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
3 pw2eng 7177 . . 3  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
42, 3syl 16 . 2  |-  ( A 
~~  B  ->  ~P A  ~~  ( 2o  ^m  A ) )
5 2onn 6846 . . . . . 6  |-  2o  e.  om
65elexi 2929 . . . . 5  |-  2o  e.  _V
76enref 7103 . . . 4  |-  2o  ~~  2o
8 mapen 7234 . . . 4  |-  ( ( 2o  ~~  2o  /\  A  ~~  B )  -> 
( 2o  ^m  A
)  ~~  ( 2o  ^m  B ) )
97, 8mpan 652 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ( 2o  ^m  B ) )
101brrelex2i 4882 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
11 pw2eng 7177 . . . 4  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
12 ensym 7119 . . . 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 7122 . . 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 7122 . 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 1721   _Vcvv 2920   ~Pcpw 3763   class class class wbr 4176   omcom 4808  (class class class)co 6044   2oc2o 6681    ^m cmap 6981    ~~ cen 7069
This theorem is referenced by:  pwfi  7364  dfac12k  7987  pwcdaidm  8035  pwsdompw  8044  ackbij2lem2  8080  engch  8463  gchdomtri  8464  canthp1lem1  8487  gchcdaidm  8503  gchxpidm  8504  gchhar  8506  gchpwdom  8509  inar1  8610  rexpen  12786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-1o 6687  df-2o 6688  df-er 6868  df-map 6983  df-en 7073
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