iccordt - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iccordt		Structured version Visualization version GIF version

Theorem iccordt 23162

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 7422	. 2 ⊢ (𝐴[,]𝐵) = ([,]‘⟨𝐴, 𝐵⟩)
2		letsr 18588	. . . . . 6 ⊢ ≤ ∈ TosetRel
3		ledm 18585	. . . . . . 7 ⊢ ℝ^* = dom ≤
4	3	ordtcld3 23147	. . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
5	2, 4	mp3an1 1444	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )))
6	5	rgen2 3187	. . . 4 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))
7		df-icc 13366	. . . . 5 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
8	7	fmpo 8073	. . . 4 ⊢ (∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) ↔ [,]:(ℝ^* × ℝ^*)⟶(Clsd‘(ordTop‘ ≤ )))
9	6, 8	mpbi 229	. . 3 ⊢ [,]:(ℝ^* × ℝ^*)⟶(Clsd‘(ordTop‘ ≤ ))
10		letop 23154	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top
11		0cld 22986	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ )))
12	10, 11	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ ))
13	9, 12	f0cli 7107	. 2 ⊢ ([,]‘⟨𝐴, 𝐵⟩) ∈ (Clsd‘(ordTop‘ ≤ ))
14	1, 13	eqeltri 2821	1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 ∈ wcel 2098 ∀wral 3050 {crab 3418 ∅c0 4322 ⟨cop 4636 class class class wbr 5149 × cxp 5676 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝ^*cxr 11279 ≤ cle 11281 [,]cicc 13362 ordTopcordt 17484 TosetRel ctsr 18560 Topctop 22839 Clsdccld 22964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-icc 13366 df-topgen 17428 df-ordt 17486 df-ps 18561 df-tsr 18562 df-top 22840 df-topon 22857 df-bases 22893 df-cld 22967

This theorem is referenced by: lecldbas 23167 icccldii 48120

W3C validator