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| 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 7435 | . 2 ⊢ (𝐴[,]𝐵) = ([,]‘〈𝐴, 𝐵〉) | |
| 2 | letsr 18639 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
| 3 | ledm 18636 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
| 4 | 3 | ordtcld3 23208 | . . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) | 
| 5 | 2, 4 | mp3an1 1449 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) | 
| 6 | 5 | rgen2 3198 | . . . 4 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) | 
| 7 | df-icc 13395 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
| 8 | 7 | fmpo 8094 | . . . 4 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) ↔ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ ))) | 
| 9 | 6, 8 | mpbi 230 | . . 3 ⊢ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ )) | 
| 10 | letop 23215 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
| 11 | 0cld 23047 | . . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ ))) | |
| 12 | 10, 11 | ax-mp 5 | . . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ )) | 
| 13 | 9, 12 | f0cli 7117 | . 2 ⊢ ([,]‘〈𝐴, 𝐵〉) ∈ (Clsd‘(ordTop‘ ≤ )) | 
| 14 | 1, 13 | eqeltri 2836 | 1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∈ wcel 2107 ∀wral 3060 {crab 3435 ∅c0 4332 〈cop 4631 class class class wbr 5142 × cxp 5682 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℝ*cxr 11295 ≤ cle 11297 [,]cicc 13391 ordTopcordt 17545 TosetRel ctsr 18611 Topctop 22900 Clsdccld 23025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-1o 8507 df-2o 8508 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-icc 13395 df-topgen 17489 df-ordt 17547 df-ps 18612 df-tsr 18613 df-top 22901 df-topon 22918 df-bases 22954 df-cld 23028 | 
| This theorem is referenced by: lecldbas 23228 icccldii 48823 | 
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