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Mirrors > Home > MPE Home > Th. List > fzp1nel | Structured version Visualization version GIF version |
Description: One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.) |
Ref | Expression |
---|---|
fzp1nel | ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11979 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | ltp1 11474 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → 𝑁 < (𝑁 + 1)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ) | |
4 | peano2re 10807 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
5 | 3, 4 | ltnled 10781 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
6 | 2, 5 | mpbid 234 | . . . . 5 ⊢ (𝑁 ∈ ℝ → ¬ (𝑁 + 1) ≤ 𝑁) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℤ → ¬ (𝑁 + 1) ≤ 𝑁) |
8 | 7 | intnand 491 | . . 3 ⊢ (𝑁 ∈ ℤ → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
9 | 8 | 3ad2ant2 1130 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
10 | elfz2 12893 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) | |
11 | 10 | notbii 322 | . . 3 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
12 | imnan 402 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) ↔ ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) | |
13 | 11, 12 | bitr4i 280 | . 2 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
14 | 9, 13 | mpbir 233 | 1 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 ℤcz 11975 ...cfz 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-z 11976 df-fz 12887 |
This theorem is referenced by: fprodm1 15315 gsumzaddlem 19035 wlkp1lem1 27449 wlkp1lem5 27453 fwddifnp1 33621 caratheodorylem1 42801 |
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