Theorem pw2en 9052

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 9051	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴))
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴)

Syntax hints:   ∈ wcel 2109  Vcvv 3450  𝒫 cpw 4565   class class class wbr 5109  (class class class)co 7389  2_oc2o 8430   ↑_m cmap 8801   ≈ cen 8917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1o 8436  df-2o 8437  df-map 8803  df-en 8921

This theorem is referenced by:  aleph1  10530  alephexp1  10538  pwcfsdom  10542  cfpwsdom  10543  hashpw  14407  rpnnen  16201  rexpen  16202