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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 6467 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
2 | eloni 6396 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordirr 6404 | . . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴) |
5 | eleq2 2828 | . . . . . . . . 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 2945 | . . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴) |
10 | onunisuc 6496 | . . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | |
11 | 9, 10 | neeqtrrd 3013 | . . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴) |
12 | 11 | neneqd 2943 | . . 3 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = ∪ suc 𝐴) |
13 | 12 | intn3an3d 1480 | . 2 ⊢ (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴)) |
14 | dflim2 6443 | . 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 1537 ∈ wcel 2106 ∅c0 4339 ∪ cuni 4912 Ord word 6385 Oncon0 6386 Lim wlim 6387 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: nlim1NEW 43432 nlim2NEW 43433 nlim3 43434 nlim4 43435 dfsucon 43513 |
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