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Mirrors > Home > MPE Home > Th. List > nnlim | Structured version Visualization version GIF version |
Description: A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
nnlim | ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord 7811 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | ordirr 6336 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
4 | elom 7806 | . . . 4 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
5 | 4 | simprbi 498 | . . 3 ⊢ (𝐴 ∈ ω → ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
6 | limeq 6330 | . . . . 5 ⊢ (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴)) | |
7 | eleq2 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 345 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
9 | 8 | spcgv 3554 | . . 3 ⊢ (𝐴 ∈ ω → (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
10 | 5, 9 | mpd 15 | . 2 ⊢ (𝐴 ∈ ω → (Lim 𝐴 → 𝐴 ∈ 𝐴)) |
11 | 3, 10 | mtod 197 | 1 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 = wceq 1542 ∈ wcel 2107 Ord word 6317 Oncon0 6318 Lim wlim 6319 ωcom 7803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-ord 6321 df-on 6322 df-lim 6323 df-om 7804 |
This theorem is referenced by: omssnlim 7818 nnsuc 7821 cantnfp1lem2 9620 cantnflem1 9630 cnfcom2lem 9642 1oequni2o 35885 finxp1o 35909 finxpreclem4 35911 dflim5 41707 |
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