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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 12514 | . . . . . 6 ⊢ ℤ ⊆ ℝ | |
2 | 1 | sseli 3944 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)) |
4 | 3 | anim1d 612 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥) → (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) |
5 | eluz1 12775 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥))) | |
6 | zre 12511 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | elicopnf 13371 | . . . 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 294 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ (𝑀[,)+∞))) |
10 | 9 | ssrdv 3954 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3914 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 +∞cpnf 11194 ≤ cle 11198 ℤcz 12507 ℤ≥cuz 12771 [,)cico 13275 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-neg 11396 df-z 12508 df-uz 12772 df-ico 13279 |
This theorem is referenced by: chtvalz 33306 |
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