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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 11833 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | ltp1 11328 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → 𝑁 < (𝑁 + 1)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ) | |
4 | peano2re 10660 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
5 | 3, 4 | ltnled 10634 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
6 | 2, 5 | mpbid 233 | . . . . 5 ⊢ (𝑁 ∈ ℝ → ¬ (𝑁 + 1) ≤ 𝑁) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℤ → ¬ (𝑁 + 1) ≤ 𝑁) |
8 | 7 | intnand 489 | . . 3 ⊢ (𝑁 ∈ ℤ → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
9 | 8 | 3ad2ant2 1127 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
10 | elfz2 12749 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) | |
11 | 10 | notbii 321 | . . 3 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
12 | imnan 400 | . . 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 279 | . 2 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
14 | 9, 13 | mpbir 232 | 1 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2081 class class class wbr 4962 (class class class)co 7016 ℝcr 10382 1c1 10384 + caddc 10386 < clt 10521 ≤ cle 10522 ℤcz 11829 ...cfz 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-z 11830 df-fz 12743 |
This theorem is referenced by: fprodm1 15154 gsumzaddlem 18761 wlkp1lem1 27140 wlkp1lem5 27144 fwddifnp1 33235 caratheodorylem1 42350 |
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