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Theorem eleenn 28822
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 28817 . 2 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
21mptrcl 7017 1 (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cfv 6553  (class class class)co 7423  m cmap 8854  cr 11153  1c1 11155  cn 12259  ...cfz 13533  𝔼cee 28814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5687  df-rel 5688  df-cnv 5689  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fv 6561  df-ee 28817
This theorem is referenced by:  eleei  28823  eedimeq  28824  brbtwn  28825  brcgr  28826  eleesub  28837  eleesubd  28838  axsegconlem1  28843  axsegconlem8  28850  axeuclidlem  28888  brsegle  35880
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