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Theorem eleenn 26674
 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 26669 . 2 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
21mptrcl 6770 1 (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6348  (class class class)co 7148   ↑m cmap 8398  ℝcr 10528  1c1 10530  ℕcn 11630  ...cfz 12884  𝔼cee 26666 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ee 26669 This theorem is referenced by:  eleei  26675  eedimeq  26676  brbtwn  26677  brcgr  26678  eleesub  26689  eleesubd  26690  axsegconlem1  26695  axsegconlem8  26702  axeuclidlem  26740  brsegle  33562
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