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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 6464 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
| 2 | eloni 6393 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 3 | ordirr 6401 | . . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
| 4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴) | 
| 5 | eleq2 2829 | . . . . . . . . 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 2946 | . . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴) | 
| 10 | onunisuc 6493 | . . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | |
| 11 | 9, 10 | neeqtrrd 3014 | . . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴) | 
| 12 | 11 | neneqd 2944 | . . 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 6440 | . 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 1086 = wceq 1539 ∈ wcel 2107 ∅c0 4332 ∪ cuni 4906 Ord word 6382 Oncon0 6383 Lim wlim 6384 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 | 
| This theorem is referenced by: nlim1NEW 43460 nlim2NEW 43461 nlim3 43462 nlim4 43463 dfsucon 43541 | 
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