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Theorem pw2en 9097
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 9096 . 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 2099  Vcvv 3470  𝒫 cpw 4598   class class class wbr 5142  (class class class)co 7414  2oc2o 8474  m cmap 8838  cen 8954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1o 8480  df-2o 8481  df-map 8840  df-en 8958
This theorem is referenced by:  aleph1  10588  alephexp1  10596  pwcfsdom  10600  cfpwsdom  10601  hashpw  14421  rpnnen  16197  rexpen  16198
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