iccordt - Metamath Proof Explorer

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Theorem iccordt 23163

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 7364	. 2 ⊢ (𝐴[,]𝐵) = ([,]‘⟨𝐴, 𝐵⟩)
2		letsr 18521	. . . . . 6 ⊢ ≤ ∈ TosetRel
3		ledm 18518	. . . . . . 7 ⊢ ℝ^* = dom ≤
4	3	ordtcld3 23148	. . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
5	2, 4	mp3an1 1451	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
6	5	rgen2 3177	. . . 4 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))
7		df-icc 13273	. . . . 5 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
8	7	fmpo 8015	. . . 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 23155	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top
11		0cld 22987	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ )))
12	10, 11	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ ))
13	9, 12	f0cli 7045	. 2 ⊢ ([,]‘⟨𝐴, 𝐵⟩) ∈ (Clsd‘(ordTop‘ ≤ ))
14	1, 13	eqeltri 2833	1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∈ wcel 2114 ∀wral 3052 {crab 3400 ∅c0 4286 ⟨cop 4587 class class class wbr 5099 × cxp 5623 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ℝ^*cxr 11170 ≤ cle 11172 [,]cicc 13269 ordTopcordt 17425 TosetRel ctsr 18493 Topctop 22842 Clsdccld 22965

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-pre-lttri 11105 ax-pre-lttrn 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-1o 8400 df-2o 8401 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fi 9319 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-icc 13273 df-topgen 17368 df-ordt 17427 df-ps 18494 df-tsr 18495 df-top 22843 df-topon 22860 df-bases 22895 df-cld 22968

This theorem is referenced by: lecldbas 23168 icccldii 49241

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