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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 6909 | . 2 ⊢ (𝐴[,]𝐵) = ([,]‘〈𝐴, 𝐵〉) | |
2 | letsr 17581 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
3 | ledm 17578 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
4 | 3 | ordtcld3 21375 | . . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
5 | 2, 4 | mp3an1 1578 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
6 | 5 | rgen2a 3187 | . . . 4 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) |
7 | df-icc 12471 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
8 | 7 | fmpt2 7501 | . . . 4 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) ↔ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ ))) |
9 | 6, 8 | mpbi 222 | . . 3 ⊢ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ )) |
10 | letop 21382 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
11 | 0cld 21214 | . . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ ))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ )) |
13 | 9, 12 | f0cli 6620 | . 2 ⊢ ([,]‘〈𝐴, 𝐵〉) ∈ (Clsd‘(ordTop‘ ≤ )) |
14 | 1, 13 | eqeltri 2903 | 1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∈ wcel 2166 ∀wral 3118 {crab 3122 ∅c0 4145 〈cop 4404 class class class wbr 4874 × cxp 5341 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 ℝ*cxr 10391 ≤ cle 10393 [,]cicc 12467 ordTopcordt 16513 TosetRel ctsr 17553 Topctop 21069 Clsdccld 21192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-pre-lttri 10327 ax-pre-lttrn 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fi 8587 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-icc 12471 df-topgen 16458 df-ordt 16515 df-ps 17554 df-tsr 17555 df-top 21070 df-topon 21087 df-bases 21122 df-cld 21195 |
This theorem is referenced by: lecldbas 21395 |
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