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Mirrors > Home > MPE Home > Th. List > fzpreddisj | Structured version Visualization version GIF version |
Description: A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.) |
Ref | Expression |
---|---|
fzpreddisj | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4129 | . 2 ⊢ ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∩ {𝑀}) | |
2 | 0lt1 11378 | . . . . . . . 8 ⊢ 0 < 1 | |
3 | 0re 10859 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
4 | 1re 10857 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
5 | 3, 4 | ltnlei 10977 | . . . . . . . 8 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0) |
6 | 2, 5 | mpbi 233 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
7 | eluzel2 12467 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
8 | 7 | zred 12306 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
9 | leaddle0 11371 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑀 ↔ 1 ≤ 0)) | |
10 | 8, 4, 9 | sylancl 589 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀 + 1) ≤ 𝑀 ↔ 1 ≤ 0)) |
11 | 6, 10 | mtbiri 330 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ (𝑀 + 1) ≤ 𝑀) |
12 | 11 | intnanrd 493 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) |
13 | 12 | intnand 492 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))) |
14 | elfz2 13126 | . . . 4 ⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) ↔ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))) | |
15 | 13, 14 | sylnibr 332 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) |
16 | disjsn 4641 | . . 3 ⊢ ((((𝑀 + 1)...𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) | |
17 | 15, 16 | sylibr 237 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑀 + 1)...𝑁) ∩ {𝑀}) = ∅) |
18 | 1, 17 | eqtrid 2790 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ∩ cin 3879 ∅c0 4251 {csn 4555 class class class wbr 5067 ‘cfv 6397 (class class class)co 7231 ℝcr 10752 0cc0 10753 1c1 10754 + caddc 10756 < clt 10891 ≤ cle 10892 ℤcz 12200 ℤ≥cuz 12462 ...cfz 13119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-po 5482 df-so 5483 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-1st 7779 df-2nd 7780 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-z 12201 df-uz 12463 df-fz 13120 |
This theorem is referenced by: gsummptfzsplitl 19342 chtvalz 32345 |
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