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Theorem pw2en 6937
Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
Hypothesis
Ref Expression
pw2en.1  |-  A  e. 
_V
Assertion
Ref Expression
pw2en  |-  ~P A  ~~  ( 2o  ^m  A
)

Proof of Theorem pw2en
StepHypRef Expression
1 pw2en.1 . 2  |-  A  e. 
_V
2 pw2eng 6936 . 2  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
31, 2ax-mp 10 1  |-  ~P A  ~~  ( 2o  ^m  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2763   ~Pcpw 3599   class class class wbr 3997  (class class class)co 5792   2oc2o 6441    ^m cmap 6740    ~~ cen 6828
This theorem is referenced by:  pwcdaen  7779  ackbij1lem5  7818  aleph1  8161  alephexp1  8169  pwcfsdom  8173  cfpwsdom  8174  hashpw  11353  rpnnen  12467  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-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-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-suc 4370  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-1o 6447  df-2o 6448  df-map 6742  df-en 6832
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