![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ioossico | Structured version Visualization version GIF version |
Description: An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.) |
Ref | Expression |
---|---|
ioossico | ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioo 12217 | . 2 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-ico 12219 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
3 | xrltle 12020 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤)) | |
4 | idd 24 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 < 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 12227 | 1 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 2030 ⊆ wss 3607 class class class wbr 4685 (class class class)co 6690 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 (,)cioo 12213 [,)cico 12215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-pre-lttri 10048 ax-pre-lttrn 10049 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-ioo 12217 df-ico 12219 |
This theorem is referenced by: elicoelioo 29668 esumdivc 30273 omssubadd 30490 rpsqrtcn 30799 icomnfinre 40097 uzubico 40113 uzubico2 40115 limcresioolb 40193 icocncflimc 40420 fourierdlem41 40683 fourierdlem46 40687 fouriersw 40766 ovolval5lem3 41189 ioosshoi 41204 vonioolem2 41216 amgmwlem 42876 |
Copyright terms: Public domain | W3C validator |