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Mirrors > Home > MPE Home > Th. List > iccordt | Structured version Visualization version GIF version |
Description: A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
iccordt | ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . 2 ⊢ (𝐴[,]𝐵) = ([,]‘〈𝐴, 𝐵〉) | |
2 | letsr 17829 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
3 | ledm 17826 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
4 | 3 | ordtcld3 21804 | . . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
5 | 2, 4 | mp3an1 1445 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
6 | 5 | rgen2 3168 | . . . 4 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) |
7 | df-icc 12733 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
8 | 7 | fmpo 7748 | . . . 4 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) ↔ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ ))) |
9 | 6, 8 | mpbi 233 | . . 3 ⊢ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ )) |
10 | letop 21811 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
11 | 0cld 21643 | . . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ ))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ )) |
13 | 9, 12 | f0cli 6841 | . 2 ⊢ ([,]‘〈𝐴, 𝐵〉) ∈ (Clsd‘(ordTop‘ ≤ )) |
14 | 1, 13 | eqeltri 2886 | 1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∅c0 4243 〈cop 4531 class class class wbr 5030 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝ*cxr 10663 ≤ cle 10665 [,]cicc 12729 ordTopcordt 16764 TosetRel ctsr 17801 Topctop 21498 Clsdccld 21621 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-icc 12733 df-topgen 16709 df-ordt 16766 df-ps 17802 df-tsr 17803 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 |
This theorem is referenced by: lecldbas 21824 |
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