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Theorem pw2en 6965
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 6964 . 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 1685   _Vcvv 2790   ~Pcpw 3627   class class class wbr 4025  (class class class)co 5820   2oc2o 6469    ^m cmap 6768    ~~ cen 6856
This theorem is referenced by:  pwcdaen  7807  ackbij1lem5  7846  aleph1  8189  alephexp1  8197  pwcfsdom  8201  cfpwsdom  8202  hashpw  11383  rpnnen  12500  rexpen  12501
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1o 6475  df-2o 6476  df-map 6770  df-en 6860
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