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Theorem eleenn 25997
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 n0i 4068 . 2 (𝐴 ∈ (𝔼‘𝑁) → ¬ (𝔼‘𝑁) = ∅)
2 ovex 6823 . . . . 5 (ℝ ↑𝑚 (1...𝑛)) ∈ V
3 df-ee 25992 . . . . 5 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
42, 3dmmpti 6163 . . . 4 dom 𝔼 = ℕ
54eleq2i 2842 . . 3 (𝑁 ∈ dom 𝔼 ↔ 𝑁 ∈ ℕ)
6 ndmfv 6359 . . 3 𝑁 ∈ dom 𝔼 → (𝔼‘𝑁) = ∅)
75, 6sylnbir 320 . 2 𝑁 ∈ ℕ → (𝔼‘𝑁) = ∅)
81, 7nsyl2 144 1 (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  c0 4063  dom cdm 5249  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  cr 10137  1c1 10139  cn 11222  ...cfz 12533  𝔼cee 25989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796  df-ee 25992
This theorem is referenced by:  eleei  25998  eedimeq  25999  brbtwn  26000  brcgr  26001  eleesub  26012  eleesubd  26013  axsegconlem1  26018  axsegconlem8  26025  axeuclidlem  26063  brsegle  32552
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