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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 7895 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
| 2 | ordirr 6402 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
| 4 | elom 7890 | . . . 4 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
| 5 | 4 | simprbi 496 | . . 3 ⊢ (𝐴 ∈ ω → ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
| 6 | limeq 6396 | . . . . 5 ⊢ (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴)) | |
| 7 | eleq2 2830 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) | |
| 8 | 6, 7 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
| 9 | 8 | spcgv 3596 | . . 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 198 | 1 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Ord word 6383 Oncon0 6384 Lim wlim 6385 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-om 7888 |
| This theorem is referenced by: omssnlim 7902 nnsuc 7905 cantnfp1lem2 9719 cantnflem1 9729 cnfcom2lem 9741 1oequni2o 37369 finxp1o 37393 finxpreclem4 37395 dflim5 43342 |
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