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Mirrors > Home > MPE Home > Th. List > zle0orge1 | Structured version Visualization version GIF version |
Description: There is no integer in the open unit interval, i.e., an integer is either less than or equal to 0 or greater than or equal to 1. (Contributed by AV, 4-Jun-2023.) |
Ref | Expression |
---|---|
zle0orge1 | ⊢ (𝑍 ∈ ℤ → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn 12579 | . 2 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℝ ∧ (𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ0))) | |
2 | nnge1 12245 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 1 ≤ 𝑍) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝑍 ∈ ℝ → (𝑍 ∈ ℕ → 1 ≤ 𝑍)) |
4 | elnn0z 12576 | . . . . . 6 ⊢ (-𝑍 ∈ ℕ0 ↔ (-𝑍 ∈ ℤ ∧ 0 ≤ -𝑍)) | |
5 | le0neg1 11727 | . . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 ≤ 0 ↔ 0 ≤ -𝑍)) | |
6 | 5 | biimprd 247 | . . . . . . 7 ⊢ (𝑍 ∈ ℝ → (0 ≤ -𝑍 → 𝑍 ≤ 0)) |
7 | 6 | adantld 490 | . . . . . 6 ⊢ (𝑍 ∈ ℝ → ((-𝑍 ∈ ℤ ∧ 0 ≤ -𝑍) → 𝑍 ≤ 0)) |
8 | 4, 7 | biimtrid 241 | . . . . 5 ⊢ (𝑍 ∈ ℝ → (-𝑍 ∈ ℕ0 → 𝑍 ≤ 0)) |
9 | 3, 8 | orim12d 962 | . . . 4 ⊢ (𝑍 ∈ ℝ → ((𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ0) → (1 ≤ 𝑍 ∨ 𝑍 ≤ 0))) |
10 | 9 | imp 406 | . . 3 ⊢ ((𝑍 ∈ ℝ ∧ (𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ0)) → (1 ≤ 𝑍 ∨ 𝑍 ≤ 0)) |
11 | 10 | orcomd 868 | . 2 ⊢ ((𝑍 ∈ ℝ ∧ (𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ0)) → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍)) |
12 | 1, 11 | sylbi 216 | 1 ⊢ (𝑍 ∈ ℤ → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 ∈ wcel 2105 class class class wbr 5148 ℝcr 11113 0cc0 11114 1c1 11115 ≤ cle 11254 -cneg 11450 ℕcn 12217 ℕ0cn0 12477 ℤcz 12563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 |
This theorem is referenced by: 2mulprm 16635 |
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