iccordt - Metamath Proof Explorer

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Theorem iccordt 23157

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 7361	. 2 ⊢ (𝐴[,]𝐵) = ([,]‘⟨𝐴, 𝐵⟩)
2		letsr 18517	. . . . . 6 ⊢ ≤ ∈ TosetRel
3		ledm 18514	. . . . . . 7 ⊢ ℝ^* = dom ≤
4	3	ordtcld3 23142	. . . . . 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 3178	. . . 4 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))
7		df-icc 13269	. . . . 5 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
8	7	fmpo 8012	. . . 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 23149	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top
11		0cld 22981	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ )))
12	10, 11	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ ))
13	9, 12	f0cli 7042	. 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 3390 ∅c0 4274 ⟨cop 4574 class class class wbr 5086 × cxp 5620 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ℝ^*cxr 11166 ≤ cle 11168 [,]cicc 13265 ordTopcordt 17421 TosetRel ctsr 18489 Topctop 22836 Clsdccld 22959

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 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-pre-lttri 11101 ax-pre-lttrn 11102

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 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-1o 8396 df-2o 8397 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9315 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-icc 13269 df-topgen 17364 df-ordt 17423 df-ps 18490 df-tsr 18491 df-top 22837 df-topon 22854 df-bases 22889 df-cld 22962

This theorem is referenced by: lecldbas 23162 icccldii 49352

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