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Theorem iccordt 23243

Description: A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
iccordt	⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Proof of Theorem iccordt Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7451	. 2 ⊢ (𝐴[,]𝐵) = ([,]‘⟨𝐴, 𝐵⟩)
2		letsr 18663	. . . . . 6 ⊢ ≤ ∈ TosetRel
3		ledm 18660	. . . . . . 7 ⊢ ℝ^* = dom ≤
4	3	ordtcld3 23228	. . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
5	2, 4	mp3an1 1448	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
6	5	rgen2 3205	. . . 4 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))
7		df-icc 13414	. . . . 5 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
8	7	fmpo 8109	. . . 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 23235	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top
11		0cld 23067	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ )))
12	10, 11	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ ))
13	9, 12	f0cli 7132	. 2 ⊢ ([,]‘⟨𝐴, 𝐵⟩) ∈ (Clsd‘(ordTop‘ ≤ ))
14	1, 13	eqeltri 2840	1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∈ wcel 2108 ∀wral 3067 {crab 3443 ∅c0 4352 ⟨cop 4654 class class class wbr 5166 × cxp 5698 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℝ^*cxr 11323 ≤ cle 11325 [,]cicc 13410 ordTopcordt 17559 TosetRel ctsr 18635 Topctop 22920 Clsdccld 23045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-1o 8522 df-2o 8523 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-icc 13414 df-topgen 17503 df-ordt 17561 df-ps 18636 df-tsr 18637 df-top 22921 df-topon 22938 df-bases 22974 df-cld 23048

This theorem is referenced by: lecldbas 23248 icccldii 48598

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