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Theorem pw2en 7952
 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 7951 . 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 1977  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583  (class class class)co 6549  2𝑜c2o 7441   ↑𝑚 cmap 7744   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-map 7746  df-en 7842 This theorem is referenced by:  pwcdaen  8890  ackbij1lem5  8929  aleph1  9272  alephexp1  9280  pwcfsdom  9284  cfpwsdom  9285  hashpw  13083  rpnnen  14795  rexpen  14796
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