Theorem pw2en 9118

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 9117	. 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 3478  𝒫 cpw 4605   class class class wbr 5148  (class class class)co 7431  2_oc2o 8499   ↑_m cmap 8865   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-map 8867  df-en 8985

This theorem is referenced by:  aleph1  10609  alephexp1  10617  pwcfsdom  10621  cfpwsdom  10622  hashpw  14472  rpnnen  16260  rexpen  16261