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Theorem pw2en 8222
 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 8221 . 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 2144  Vcvv 3349  𝒫 cpw 4295   class class class wbr 4784  (class class class)co 6792  2𝑜c2o 7706   ↑𝑚 cmap 8008   ≈ cen 8105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1o 7712  df-2o 7713  df-map 8010  df-en 8109 This theorem is referenced by:  pwcdaen  9208  aleph1  9594  alephexp1  9602  pwcfsdom  9606  cfpwsdom  9607  hashpw  13424  rpnnen  15161  rexpen  15162
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