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Theorem pw2en 7206
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 7205 . 2  |-  ( A  e.  _V  ->  ~P A  ~~  ( 2o  ^m  A ) )
31, 2ax-mp 8 1  |-  ~P A  ~~  ( 2o  ^m  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204  (class class class)co 6072   2oc2o 6709    ^m cmap 7009    ~~ cen 7097
This theorem is referenced by:  pwcdaen  8054  ackbij1lem5  8093  aleph1  8435  alephexp1  8443  pwcfsdom  8447  cfpwsdom  8448  hashpw  11687  rpnnen  12814  rexpen  12815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1o 6715  df-2o 6716  df-map 7011  df-en 7101
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