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Theorem pw2en 7057
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 7056 . 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 1710   _Vcvv 2864   ~Pcpw 3701   class class class wbr 4104  (class class class)co 5945   2oc2o 6560    ^m cmap 6860    ~~ cen 6948
This theorem is referenced by:  pwcdaen  7901  ackbij1lem5  7940  aleph1  8283  alephexp1  8291  pwcfsdom  8295  cfpwsdom  8296  hashpw  11484  rpnnen  12602  rexpen  12603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1o 6566  df-2o 6567  df-map 6862  df-en 6952
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