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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 7311 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | ordirr 5963 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
4 | elom 7306 | . . . 4 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
5 | 4 | simprbi 491 | . . 3 ⊢ (𝐴 ∈ ω → ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
6 | limeq 5957 | . . . . 5 ⊢ (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴)) | |
7 | eleq2 2871 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 336 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
9 | 8 | spcgv 3485 | . . 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 190 | 1 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1651 = wceq 1653 ∈ wcel 2157 Ord word 5944 Oncon0 5945 Lim wlim 5946 ωcom 7303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-sep 4979 ax-nul 4987 ax-pr 5101 ax-un 7187 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-ral 3098 df-rex 3099 df-rab 3102 df-v 3391 df-sbc 3638 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-pss 3789 df-nul 4120 df-if 4282 df-sn 4373 df-pr 4375 df-tp 4377 df-op 4379 df-uni 4633 df-br 4848 df-opab 4910 df-tr 4950 df-eprel 5229 df-po 5237 df-so 5238 df-fr 5275 df-we 5277 df-ord 5948 df-on 5949 df-lim 5950 df-suc 5951 df-om 7304 |
This theorem is referenced by: omssnlim 7317 nnsuc 7320 cantnfp1lem2 8830 cantnflem1 8840 cnfcom2lem 8852 1oequni2o 33718 finxp1o 33731 finxpreclem4 33733 |
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