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Mirrors > Home > MPE Home > Th. List > iocssicc | Structured version Visualization version GIF version |
Description: A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
Ref | Expression |
---|---|
iocssicc | ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioc 12737 | . 2 ⊢ (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥 ∧ 𝑥 ≤ 𝑏)}) | |
2 | df-icc 12739 | . 2 ⊢ [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 ≤ 𝑏)}) | |
3 | xrltle 12536 | . 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 12746 | 1 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 (,]cioc 12733 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioc 12737 df-icc 12739 |
This theorem is referenced by: xrge0iifcnv 31171 xrge0iifcv 31172 xrge0iifhom 31175 pnfneige0 31189 lmxrge0 31190 eliccelioc 41790 limcicciooub 41911 fourierdlem17 42403 fourierdlem35 42421 fourierdlem41 42427 fourierdlem48 42433 fourierdlem49 42434 fourierdlem51 42436 fourierdlem71 42456 fourierdlem102 42487 fourierdlem114 42499 |
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