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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 12617 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 2 | ltp1 12107 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → 𝑁 < (𝑁 + 1)) | |
| 3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ) | |
| 4 | peano2re 11434 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
| 5 | 3, 4 | ltnled 11408 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
| 6 | 2, 5 | mpbid 232 | . . . . 5 ⊢ (𝑁 ∈ ℝ → ¬ (𝑁 + 1) ≤ 𝑁) |
| 7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℤ → ¬ (𝑁 + 1) ≤ 𝑁) |
| 8 | 7 | intnand 488 | . . 3 ⊢ (𝑁 ∈ ℤ → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
| 9 | 8 | 3ad2ant2 1135 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁)) |
| 10 | elfz2 13554 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) | |
| 11 | 10 | notbii 320 | . . 3 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
| 12 | imnan 399 | . . 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 278 | . 2 ⊢ (¬ (𝑁 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ¬ (𝑀 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ 𝑁))) |
| 14 | 9, 13 | mpbir 231 | 1 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 ℤcz 12613 ...cfz 13547 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-z 12614 df-fz 13548 |
| This theorem is referenced by: fprodm1 16003 gsumzaddlem 19939 wlkp1lem1 29691 wlkp1lem5 29695 fwddifnp1 36166 caratheodorylem1 46541 |
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