![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 12564 | . . . . . 6 ⊢ ℤ ⊆ ℝ | |
2 | 1 | sseli 3978 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)) |
4 | 3 | anim1d 611 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥) → (𝑥 ∈ ℝ ∧ 𝑀 ≤ 𝑥))) |
5 | eluz1 12825 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ (ℤ≥‘𝑀) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥))) | |
6 | zre 12561 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | elicopnf 13421 | . . . 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 3988 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 +∞cpnf 11244 ≤ cle 11248 ℤcz 12557 ℤ≥cuz 12821 [,)cico 13325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-neg 11446 df-z 12558 df-uz 12822 df-ico 13329 |
This theorem is referenced by: chtvalz 33636 |
Copyright terms: Public domain | W3C validator |