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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 4196 | . 2 ⊢ ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∩ {𝑀}) | |
2 | 0lt1 11740 | . . . . . . . 8 ⊢ 0 < 1 | |
3 | 0re 11220 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
4 | 1re 11218 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
5 | 3, 4 | ltnlei 11339 | . . . . . . . 8 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0) |
6 | 2, 5 | mpbi 229 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
7 | eluzel2 12831 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
8 | 7 | zred 12670 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
9 | leaddle0 11733 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑀 ↔ 1 ≤ 0)) | |
10 | 8, 4, 9 | sylancl 585 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀 + 1) ≤ 𝑀 ↔ 1 ≤ 0)) |
11 | 6, 10 | mtbiri 327 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ (𝑀 + 1) ≤ 𝑀) |
12 | 11 | intnanrd 489 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) |
13 | 12 | intnand 488 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))) |
14 | elfz2 13497 | . . . 4 ⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) ↔ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑀 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))) | |
15 | 13, 14 | sylnibr 329 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) |
16 | disjsn 4710 | . . 3 ⊢ ((((𝑀 + 1)...𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) | |
17 | 15, 16 | sylibr 233 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑀 + 1)...𝑁) ∩ {𝑀}) = ∅) |
18 | 1, 17 | eqtrid 2778 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ∅c0 4317 {csn 4623 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 ℤcz 12562 ℤ≥cuz 12826 ...cfz 13490 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13491 |
This theorem is referenced by: gsummptfzsplitl 19853 chtvalz 34170 |
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