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Theorem pw2en 8610
 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 8609 . 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 2111  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5031  (class class class)co 7136  2oc2o 8082   ↑m cmap 8392   ≈ cen 8492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1o 8088  df-2o 8089  df-map 8394  df-en 8496 This theorem is referenced by:  aleph1  9985  alephexp1  9993  pwcfsdom  9997  cfpwsdom  9998  hashpw  13796  rpnnen  15575  rexpen  15576
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