iooordt - Metamath Proof Explorer

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Theorem iooordt 22713

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

**Assertion**
Ref	Expression
iooordt	⊢ (𝐴(,)𝐵) ∈ (ordTop‘ ≤ )

**Proof of Theorem iooordt**
Step	Hyp	Ref	Expression
1		eqid 2733	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		eqid 2733	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
3		eqid 2733	. . . . . . 7 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 22709	. . . . . 6 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
5		letop 22702	. . . . . 6 ⊢ (ordTop‘ ≤ ) ∈ Top
6	4, 5	eqeltrri 2831	. . . . 5 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
7		tgclb 22465	. . . . 5 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases ↔ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top)
8	6, 7	mpbir 230	. . . 4 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases
9		bastg 22461	. . . 4 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases → ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))))
10	8, 9	ax-mp 5	. . 3 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
11	10, 4	sseqtrri 4019	. 2 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (ordTop‘ ≤ )
12		ssun2 4173	. . 3 ⊢ ran (,) ⊆ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))
13		ioorebas 13425	. . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,)
14	12, 13	sselii 3979	. 2 ⊢ (𝐴(,)𝐵) ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))
15	11, 14	sselii 3979	1 ⊢ (𝐴(,)𝐵) ∈ (ordTop‘ ≤ )

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2107 ∪ cun 3946 ⊆ wss 3948 ↦ cmpt 5231 ran crn 5677 ‘cfv 6541 (class class class)co 7406 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 ≤ cle 11246 (,)cioo 13321 (,]cioc 13322 [,)cico 13323 topGenctg 17380 ordTopcordt 17442 Topctop 22387 TopBasesctb 22440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-topgen 17386 df-ordt 17444 df-ps 18516 df-tsr 18517 df-top 22388 df-topon 22405 df-bases 22441

This theorem is referenced by: reordt 22714 xrtgioo 24314 xlimxrre 44534

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