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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 6965 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | ordirr 5658 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
4 | elom 6960 | . . . 4 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
5 | 4 | simprbi 479 | . . 3 ⊢ (𝐴 ∈ ω → ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
6 | limeq 5652 | . . . . 5 ⊢ (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴)) | |
7 | eleq2 2677 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 333 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
9 | 8 | spcgv 3266 | . . 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 188 | 1 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Ord word 5639 Oncon0 5640 Lim wlim 5641 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: omssnlim 6971 nnsuc 6974 cantnfp1lem2 8459 cantnflem1 8469 cnfcom2lem 8481 1oequni2o 32392 finxp1o 32405 finxpreclem4 32407 |
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