MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwen Unicode version

Theorem pwen 7209
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 7043 . . . 4  |-  Rel  ~~
21brrelexi 4851 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
3 pw2eng 7143 . . 3  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
42, 3syl 16 . 2  |-  ( A 
~~  B  ->  ~P A  ~~  ( 2o  ^m  A ) )
5 2onn 6812 . . . . . 6  |-  2o  e.  om
65elexi 2901 . . . . 5  |-  2o  e.  _V
76enref 7069 . . . 4  |-  2o  ~~  2o
8 mapen 7200 . . . 4  |-  ( ( 2o  ~~  2o  /\  A  ~~  B )  -> 
( 2o  ^m  A
)  ~~  ( 2o  ^m  B ) )
97, 8mpan 652 . . 3  |-  ( A 
~~  B  ->  ( 2o  ^m  A )  ~~  ( 2o  ^m  B ) )
101brrelex2i 4852 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
11 pw2eng 7143 . . . 4  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
12 ensym 7085 . . . 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 7088 . . 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 7088 . 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 1717   _Vcvv 2892   ~Pcpw 3735   class class class wbr 4146   omcom 4778  (class class class)co 6013   2oc2o 6647    ^m cmap 6947    ~~ cen 7035
This theorem is referenced by:  pwfi  7330  dfac12k  7953  pwcdaidm  8001  pwsdompw  8010  ackbij2lem2  8046  engch  8429  gchdomtri  8430  canthp1lem1  8453  gchcdaidm  8469  gchxpidm  8470  gchhar  8472  gchpwdom  8475  inar1  8576  rexpen  12747
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-1o 6653  df-2o 6654  df-er 6834  df-map 6949  df-en 7039
  Copyright terms: Public domain W3C validator