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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 7434 | . 2 ⊢ (𝐴[,]𝐵) = ([,]‘〈𝐴, 𝐵〉) | |
2 | letsr 18651 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
3 | ledm 18648 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
4 | 3 | ordtcld3 23223 | . . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
5 | 2, 4 | mp3an1 1447 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
6 | 5 | rgen2 3197 | . . . 4 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) |
7 | df-icc 13391 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
8 | 7 | fmpo 8092 | . . . 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 23230 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
11 | 0cld 23062 | . . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ ))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ )) |
13 | 9, 12 | f0cli 7118 | . 2 ⊢ ([,]‘〈𝐴, 𝐵〉) ∈ (Clsd‘(ordTop‘ ≤ )) |
14 | 1, 13 | eqeltri 2835 | 1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∅c0 4339 〈cop 4637 class class class wbr 5148 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝ*cxr 11292 ≤ cle 11294 [,]cicc 13387 ordTopcordt 17546 TosetRel ctsr 18623 Topctop 22915 Clsdccld 23040 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-icc 13391 df-topgen 17490 df-ordt 17548 df-ps 18624 df-tsr 18625 df-top 22916 df-topon 22933 df-bases 22969 df-cld 23043 |
This theorem is referenced by: lecldbas 23243 icccldii 48715 |
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