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Theorem pw2en 9119
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. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
Hypothesis
Ref Expression
pw2en.1 𝐴 ∈ V
Assertion
Ref Expression
pw2en 𝒫 𝐴 ≈ (2om 𝐴)

Proof of Theorem pw2en
StepHypRef Expression
1 pw2en.1 . 2 𝐴 ∈ V
2 pw2eng 9118 . 2 (𝐴 ∈ V → 𝒫 𝐴 ≈ (2om 𝐴))
31, 2ax-mp 5 1 𝒫 𝐴 ≈ (2om 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143  (class class class)co 7431  2oc2o 8500  m cmap 8866  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986
This theorem is referenced by:  aleph1  10611  alephexp1  10619  pwcfsdom  10623  cfpwsdom  10624  hashpw  14475  rpnnen  16263  rexpen  16264
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