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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzne1 | Structured version Visualization version GIF version |
Description: Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
Ref | Expression |
---|---|
fzne1 | ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2942 | . 2 ⊢ (𝐾 ≠ 𝑀 ↔ ¬ 𝐾 = 𝑀) | |
2 | elfzuz2 13142 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
3 | elfzp12 13216 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
5 | 4 | ibi 270 | . . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) |
6 | 5 | orcanai 1003 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ ¬ 𝐾 = 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁)) |
7 | 1, 6 | sylan2b 597 | 1 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ‘cfv 6398 (class class class)co 7232 1c1 10755 + caddc 10757 ℤ≥cuz 12463 ...cfz 13120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-n0 12116 df-z 12202 df-uz 12464 df-fz 13121 |
This theorem is referenced by: fzm1ne1 30857 cycpmco2lem4 31142 cycpmco2lem5 31143 |
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