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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 7590 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | ordirr 6211 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
4 | elom 7585 | . . . 4 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
5 | 4 | simprbi 499 | . . 3 ⊢ (𝐴 ∈ ω → ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
6 | limeq 6205 | . . . . 5 ⊢ (𝑥 = 𝐴 → (Lim 𝑥 ↔ Lim 𝐴)) | |
7 | eleq2 2903 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim 𝐴 → 𝐴 ∈ 𝐴))) |
9 | 8 | spcgv 3597 | . . 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 200 | 1 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Ord word 6192 Oncon0 6193 Lim wlim 6194 ωcom 7582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-om 7583 |
This theorem is referenced by: omssnlim 7596 nnsuc 7599 cantnfp1lem2 9144 cantnflem1 9154 cnfcom2lem 9166 1oequni2o 34651 finxp1o 34675 finxpreclem4 34677 |
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