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Theorem eleenn 28912
Description: If 𝐴 is in (𝔼‘𝑁), then 𝑁 is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
Assertion
Ref Expression
eleenn (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)

Proof of Theorem eleenn
StepHypRef Expression
1 df-ee 28907 . 2 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
21mptrcl 7024 1 (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6560  (class class class)co 7432  m cmap 8867  cr 11155  1c1 11157  cn 12267  ...cfz 13548  𝔼cee 28904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568  df-ee 28907
This theorem is referenced by:  eleei  28913  eedimeq  28914  brbtwn  28915  brcgr  28916  eleesub  28927  eleesubd  28928  axsegconlem1  28933  axsegconlem8  28940  axeuclidlem  28978  brsegle  36110
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