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Theorem nnlim 7884
Description: A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
Assertion
Ref Expression
nnlim (𝐴 ∈ ω → ¬ Lim 𝐴)

Proof of Theorem nnlim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nnord 7878 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordirr 6387 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → ¬ 𝐴𝐴)
4 elom 7873 . . . 4 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
54simprbi 496 . . 3 (𝐴 ∈ ω → ∀𝑥(Lim 𝑥𝐴𝑥))
6 limeq 6381 . . . . 5 (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴))
7 eleq2 2818 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
86, 7imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((Lim 𝑥𝐴𝑥) ↔ (Lim 𝐴𝐴𝐴)))
98spcgv 3583 . . 3 (𝐴 ∈ ω → (∀𝑥(Lim 𝑥𝐴𝑥) → (Lim 𝐴𝐴𝐴)))
105, 9mpd 15 . 2 (𝐴 ∈ ω → (Lim 𝐴𝐴𝐴))
113, 10mtod 197 1 (𝐴 ∈ ω → ¬ Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532   = wceq 1534  wcel 2099  Ord word 6368  Oncon0 6369  Lim wlim 6370  ωcom 7870
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-on 6373  df-lim 6374  df-om 7871
This theorem is referenced by:  omssnlim  7885  nnsuc  7888  cantnfp1lem2  9703  cantnflem1  9713  cnfcom2lem  9725  1oequni2o  36847  finxp1o  36871  finxpreclem4  36873  dflim5  42758
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