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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 6445 | . . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴) | |
2 | ordsuc 7832 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
3 | 1, 2 | sylibr 234 | . . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴) |
4 | limuni 6446 | . . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴) | |
5 | ordunisuc 7851 | . . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 5 | eqeq2d 2745 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴)) |
7 | ordirr 6403 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | eleq2 2827 | . . . . . . 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 6466 | . . . . . 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 1536 ∈ wcel 2105 ∪ cuni 4911 Ord word 6384 Lim wlim 6386 suc csuc 6387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 |
This theorem is referenced by: tz7.44-2 8445 rankxpsuc 9919 scutbdaybnd2lim 27876 dfrdg2 35776 dfrdg4 35932 onov0suclim 43263 dflim5 43318 |
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