![]() |
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 2994 | . 2 ⊢ (𝜑 → 𝑁 ≠ 𝐾) |
3 | fzne2d.1 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
4 | elfz2 13551 | . . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
5 | 3, 4 | sylib 218 | . . . . . 6 ⊢ (𝜑 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
6 | 5 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) |
7 | 6 | simp3d 1143 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
8 | 7 | zred 12720 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
9 | 6 | simp2d 1142 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | 9 | zred 12720 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
11 | 5 | simprrd 774 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑁) |
12 | 8, 10, 11 | leltned 11412 | . 2 ⊢ (𝜑 → (𝐾 < 𝑁 ↔ 𝑁 ≠ 𝐾)) |
13 | 2, 12 | mpbird 257 | 1 ⊢ (𝜑 → 𝐾 < 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 < clt 11293 ≤ cle 11294 ℤcz 12611 ...cfz 13544 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-fz 13545 |
This theorem is referenced by: metakunt2 42188 |
Copyright terms: Public domain | W3C validator |