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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 12743 | . 2 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-ico 12745 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
3 | xrltle 12543 | . 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 12753 | 1 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 (class class class)co 7156 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 (,)cioo 12739 [,)cico 12741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 df-ico 12745 |
This theorem is referenced by: elicoelioo 30501 esumdivc 31342 omssubadd 31558 rpsqrtcn 31864 icomnfinre 41848 uzubico 41864 uzubico2 41866 limcresioolb 41944 icocncflimc 42192 fourierdlem41 42453 fourierdlem46 42457 fouriersw 42536 ovolval5lem3 42956 ioosshoi 42971 vonioolem2 42983 amgmwlem 44923 |
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