Theorem eleenn 26195

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

Step	Hyp	Ref	Expression
1		n0i 4149	. 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → ¬ (𝔼‘𝑁) = ∅)
2		ovex 6937	. . . . 5 ⊢ (ℝ ↑𝑚 (1...𝑛)) ∈ V
3		df-ee 26190	. . . . 5 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
4	2, 3	dmmpti 6256	. . . 4 ⊢ dom 𝔼 = ℕ
5	4	eleq2i 2898	. . 3 ⊢ (𝑁 ∈ dom 𝔼 ↔ 𝑁 ∈ ℕ)
6		ndmfv 6463	. . 3 ⊢ (¬ 𝑁 ∈ dom 𝔼 → (𝔼‘𝑁) = ∅)
7	5, 6	sylnbir 323	. 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝔼‘𝑁) = ∅)
8	1, 7	nsyl2 145	1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ∅c0 4144  dom cdm 5342  ‘cfv 6123  (class class class)co 6905   ↑_𝑚 cmap 8122  ℝcr 10251  1c1 10253  ℕcn 11350  ...cfz 12619  𝔼cee 26187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-ov 6908  df-ee 26190

This theorem is referenced by:  eleei  26196  eedimeq  26197  brbtwn  26198  brcgr  26199  eleesub  26210  eleesubd  26211  axsegconlem1  26216  axsegconlem8  26223  axeuclidlem  26261  brsegle  32754