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Theorem nnlim 7032
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 7027 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordirr 5705 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → ¬ 𝐴𝐴)
4 elom 7022 . . . 4 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
54simprbi 480 . . 3 (𝐴 ∈ ω → ∀𝑥(Lim 𝑥𝐴𝑥))
6 limeq 5699 . . . . 5 (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴))
7 eleq2 2687 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
86, 7imbi12d 334 . . . 4 (𝑥 = 𝐴 → ((Lim 𝑥𝐴𝑥) ↔ (Lim 𝐴𝐴𝐴)))
98spcgv 3282 . . 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 1478   = wceq 1480  wcel 1987  Ord word 5686  Oncon0 5687  Lim wlim 5688  ωcom 7019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-om 7020
This theorem is referenced by:  omssnlim  7033  nnsuc  7036  cantnfp1lem2  8528  cantnflem1  8538  cnfcom2lem  8550  1oequni2o  32883  finxp1o  32896  finxpreclem4  32898
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