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Theorem nnlim 7120
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 7115 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordirr 5779 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → ¬ 𝐴𝐴)
4 elom 7110 . . . 4 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
54simprbi 479 . . 3 (𝐴 ∈ ω → ∀𝑥(Lim 𝑥𝐴𝑥))
6 limeq 5773 . . . . 5 (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴))
7 eleq2 2719 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
86, 7imbi12d 333 . . . 4 (𝑥 = 𝐴 → ((Lim 𝑥𝐴𝑥) ↔ (Lim 𝐴𝐴𝐴)))
98spcgv 3324 . . 3 (𝐴 ∈ ω → (∀𝑥(Lim 𝑥𝐴𝑥) → (Lim 𝐴𝐴𝐴)))
105, 9mpd 15 . 2 (𝐴 ∈ ω → (Lim 𝐴𝐴𝐴))
113, 10mtod 189 1 (𝐴 ∈ ω → ¬ Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521   = wceq 1523  wcel 2030  Ord word 5760  Oncon0 5761  Lim wlim 5762  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108
This theorem is referenced by:  omssnlim  7121  nnsuc  7124  cantnfp1lem2  8614  cantnflem1  8624  cnfcom2lem  8636  1oequni2o  33346  finxp1o  33359  finxpreclem4  33361
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