![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > icossicc | GIF version |
Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.) |
Ref | Expression |
---|---|
icossicc | ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 9912 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 9913 | . 2 ⊢ [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 ≤ 𝑏)}) | |
3 | idd 21 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 ≤ 𝑤 → 𝐴 ≤ 𝑤)) | |
4 | xrltle 9816 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 9920 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2160 ⊆ wss 3144 class class class wbr 4018 (class class class)co 5891 ℝ*cxr 8009 < clt 8010 ≤ cle 8011 [,)cico 9908 [,]cicc 9909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-pre-ltirr 7941 ax-pre-lttrn 7943 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-ico 9912 df-icc 9913 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |