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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 6418 | . . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴) | |
2 | ordsuc 7798 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
3 | 1, 2 | sylibr 233 | . . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴) |
4 | limuni 6419 | . . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴) | |
5 | ordunisuc 7817 | . . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 5 | eqeq2d 2737 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴)) |
7 | ordirr 6376 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | eleq2 2816 | . . . . . . 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 246 | . . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴)) |
11 | sucidg 6439 | . . . . . 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 239 | . . 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 1533 ∈ wcel 2098 ∪ cuni 4902 Ord word 6357 Lim wlim 6359 suc csuc 6360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 |
This theorem is referenced by: tz7.44-2 8408 rankxpsuc 9879 scutbdaybnd2lim 27705 dfrdg2 35300 dfrdg4 35456 onov0suclim 42597 dflim5 42652 |
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