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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzssico | Structured version Visualization version GIF version |
Description: Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
Ref | Expression |
---|---|
uzssico | ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssre 12601 | . . . . . 6 ⊢ ℤ ⊆ ℝ | |
2 | 1 | sseli 3976 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)) |
4 | 3 | anim1d 609 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥) → (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) |
5 | eluz1 12862 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥))) | |
6 | zre 12598 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | elicopnf 13460 | . . . 4 ⊢ (𝑀 ∈ ℝ → (𝑥 ∈ (𝑀[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (𝑀[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) |
9 | 4, 5, 8 | 3imtr4d 293 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ (𝑀[,)+∞))) |
10 | 9 | ssrdv 3986 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3947 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 ℝcr 11143 +∞cpnf 11281 ≤ cle 11285 ℤcz 12594 ℤ≥cuz 12858 [,)cico 13364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-pre-lttri 11218 ax-pre-lttrn 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-neg 11483 df-z 12595 df-uz 12859 df-ico 13368 |
This theorem is referenced by: chtvalz 34266 |
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