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Theorem pw2en 6854
 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. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
Hypothesis
Ref Expression
pw2en.1
Assertion
Ref Expression
pw2en

Proof of Theorem pw2en
StepHypRef Expression
1 pw2en.1 . 2
2 pw2eng 6853 . 2
31, 2ax-mp 10 1
 Colors of variables: wff set class Syntax hints:   wcel 1621  cvv 2727  cpw 3530   class class class wbr 3920  (class class class)co 5710  c2o 6359   cmap 6658   cen 6746 This theorem is referenced by:  pwcdaen  7695  ackbij1lem5  7734  aleph1  8073  alephexp1  8081  pwcfsdom  8085  cfpwsdom  8086  hashpw  11265  rpnnen  12379  rexpen  12380 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1o 6365  df-2o 6366  df-map 6660  df-en 6750
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