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Mirrors > Home > MPE Home > Th. List > Mathboxes > nlimsuc | Structured version Visualization version GIF version |
Description: A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.) |
Ref | Expression |
---|---|
nlimsuc | ⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 6403 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
2 | eloni 6332 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordirr 6340 | . . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴) |
5 | eleq2 2827 | . . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
6 | 5 | notbid 318 | . . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴)) |
7 | 4, 6 | syl5ibrcom 247 | . . . . . . 7 ⊢ (𝐴 ∈ On → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴)) |
8 | 1, 7 | mt2d 136 | . . . . . 6 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = 𝐴) |
9 | 8 | neqned 2951 | . . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴) |
10 | onunisuc 6432 | . . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | |
11 | 9, 10 | neeqtrrd 3019 | . . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴) |
12 | 11 | neneqd 2949 | . . 3 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = ∪ suc 𝐴) |
13 | 12 | intn3an3d 1482 | . 2 ⊢ (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴)) |
14 | dflim2 6379 | . 2 ⊢ (Lim suc 𝐴 ↔ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴)) | |
15 | 13, 14 | sylnibr 329 | 1 ⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∅c0 4287 ∪ cuni 4870 Ord word 6321 Oncon0 6322 Lim wlim 6323 suc csuc 6324 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 |
This theorem is referenced by: nlim1NEW 41788 nlim2NEW 41789 nlim3 41790 nlim4 41791 dfsucon 41869 |
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