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| Mirrors > Home > MPE Home > Th. List > nlimsucg | Structured version Visualization version GIF version | ||
| Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| nlimsucg | ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | limord 6444 | . . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴) | |
| 2 | ordsuc 7833 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
| 3 | 1, 2 | sylibr 234 | . . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴) |
| 4 | limuni 6445 | . . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴) | |
| 5 | ordunisuc 7852 | . . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
| 6 | 5 | eqeq2d 2748 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴)) |
| 7 | ordirr 6402 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
| 8 | eleq2 2830 | . . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
| 9 | 8 | notbid 318 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴)) |
| 10 | 7, 9 | syl5ibrcom 247 | . . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴)) |
| 11 | sucidg 6465 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
| 12 | 11 | con3i 154 | . . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
| 13 | 10, 12 | syl6 35 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
| 14 | 6, 13 | sylbid 240 | . . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
| 15 | 3, 4, 14 | sylc 65 | . 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
| 16 | 15 | con2i 139 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 Ord word 6383 Lim wlim 6385 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 |
| This theorem is referenced by: tz7.44-2 8447 rankxpsuc 9922 scutbdaybnd2lim 27862 dfrdg2 35796 dfrdg4 35952 onov0suclim 43287 dflim5 43342 |
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