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Theorem nnlim 7890
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 7884 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordirr 6394 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → ¬ 𝐴𝐴)
4 elom 7879 . . . 4 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
54simprbi 495 . . 3 (𝐴 ∈ ω → ∀𝑥(Lim 𝑥𝐴𝑥))
6 limeq 6388 . . . . 5 (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴))
7 eleq2 2815 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
86, 7imbi12d 343 . . . 4 (𝑥 = 𝐴 → ((Lim 𝑥𝐴𝑥) ↔ (Lim 𝐴𝐴𝐴)))
98spcgv 3582 . . 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 6375  Oncon0 6376  Lim wlim 6377  ωcom 7876
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-lim 6381  df-om 7877
This theorem is referenced by:  omssnlim  7891  nnsuc  7894  cantnfp1lem2  9722  cantnflem1  9732  cnfcom2lem  9744  1oequni2o  37075  finxp1o  37099  finxpreclem4  37101  dflim5  42995
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