![]() |
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fzne2d | Structured version Visualization version GIF version |
Description: Elementhood in a finite set of sequential integers, except its upper bound. (Contributed by metakunt, 23-May-2024.) |
Ref | Expression |
---|---|
fzne2d.1 | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
fzne2d.2 | ⊢ (𝜑 → 𝐾 ≠ 𝑁) |
Ref | Expression |
---|---|
fzne2d | ⊢ (𝜑 → 𝐾 < 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzne2d.2 | . . 3 ⊢ (𝜑 → 𝐾 ≠ 𝑁) | |
2 | 1 | necomd 3004 | . 2 ⊢ (𝜑 → 𝑁 ≠ 𝐾) |
3 | fzne2d.1 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
4 | elfz2 12931 | . . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
5 | 3, 4 | sylib 221 | . . . . . 6 ⊢ (𝜑 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
6 | 5 | simpld 499 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) |
7 | 6 | simp3d 1142 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
8 | 7 | zred 12111 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
9 | 6 | simp2d 1141 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | 9 | zred 12111 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
11 | 5 | simprrd 774 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑁) |
12 | 8, 10, 11 | leltned 10816 | . 2 ⊢ (𝜑 → (𝐾 < 𝑁 ↔ 𝑁 ≠ 𝐾)) |
13 | 2, 12 | mpbird 260 | 1 ⊢ (𝜑 → 𝐾 < 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 ∈ wcel 2112 ≠ wne 2949 class class class wbr 5025 (class class class)co 7143 < clt 10698 ≤ cle 10699 ℤcz 12005 ...cfz 12924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-pre-lttri 10634 ax-pre-lttrn 10635 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-neg 10896 df-z 12006 df-fz 12925 |
This theorem is referenced by: metakunt2 39633 |
Copyright terms: Public domain | W3C validator |