Theorem pw2en 8866

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 ≈ (2o ↑m 𝐴)

Proof of Theorem pw2en

Step	Hyp	Ref	Expression
1		pw2en.1	. 2 ⊢ 𝐴 ∈ V
2		pw2eng 8865	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴))
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴)

Syntax hints:   ∈ wcel 2106  Vcvv 3432  𝒫 cpw 4533   class class class wbr 5074  (class class class)co 7275  2_oc2o 8291   ↑_m cmap 8615   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-map 8617  df-en 8734

This theorem is referenced by:  aleph1  10327  alephexp1  10335  pwcfsdom  10339  cfpwsdom  10340  hashpw  14151  rpnnen  15936  rexpen  15937