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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 12642 | . . . . . 6 ⊢ ℤ ⊆ ℝ | |
2 | 1 | sseli 3998 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)) |
4 | 3 | anim1d 610 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥) → (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) |
5 | eluz1 12903 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥))) | |
6 | zre 12639 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | elicopnf 13501 | . . . 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 4008 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2103 ⊆ wss 3970 class class class wbr 5169 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 +∞cpnf 11317 ≤ cle 11321 ℤcz 12635 ℤ≥cuz 12899 [,)cico 13405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-pre-lttri 11254 ax-pre-lttrn 11255 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-neg 11519 df-z 12636 df-uz 12900 df-ico 13409 |
This theorem is referenced by: chtvalz 34598 |
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