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Mirrors > Home > MPE Home > Th. List > Mathboxes > iooii | Structured version Visualization version GIF version |
Description: Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.) |
Ref | Expression |
---|---|
iooii | ⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11050 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1xr 11062 | . . 3 ⊢ 1 ∈ ℝ* | |
3 | ioossioo 13201 | . . 3 ⊢ (((0 ∈ ℝ* ∧ 1 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 𝐵 ≤ 1)) → (𝐴(,)𝐵) ⊆ (0(,)1)) | |
4 | 1, 2, 3 | mpanl12 698 | . 2 ⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ⊆ (0(,)1)) |
5 | iooretop 23957 | . . . 4 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
6 | iooretop 23957 | . . . . 5 ⊢ (0(,)1) ∈ (topGen‘ran (,)) | |
7 | ioossicc 13193 | . . . . 5 ⊢ (0(,)1) ⊆ (0[,]1) | |
8 | retop 23953 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
9 | ovex 7328 | . . . . . 6 ⊢ (0[,]1) ∈ V | |
10 | restopnb 22354 | . . . . . 6 ⊢ ((((topGen‘ran (,)) ∈ Top ∧ (0[,]1) ∈ V) ∧ ((0(,)1) ∈ (topGen‘ran (,)) ∧ (0(,)1) ⊆ (0[,]1) ∧ (𝐴(,)𝐵) ⊆ (0(,)1))) → ((𝐴(,)𝐵) ∈ (topGen‘ran (,)) ↔ (𝐴(,)𝐵) ∈ ((topGen‘ran (,)) ↾t (0[,]1)))) | |
11 | 8, 9, 10 | mpanl12 698 | . . . . 5 ⊢ (((0(,)1) ∈ (topGen‘ran (,)) ∧ (0(,)1) ⊆ (0[,]1) ∧ (𝐴(,)𝐵) ⊆ (0(,)1)) → ((𝐴(,)𝐵) ∈ (topGen‘ran (,)) ↔ (𝐴(,)𝐵) ∈ ((topGen‘ran (,)) ↾t (0[,]1)))) |
12 | 6, 7, 11 | mp3an12 1449 | . . . 4 ⊢ ((𝐴(,)𝐵) ⊆ (0(,)1) → ((𝐴(,)𝐵) ∈ (topGen‘ran (,)) ↔ (𝐴(,)𝐵) ∈ ((topGen‘ran (,)) ↾t (0[,]1)))) |
13 | 5, 12 | mpbii 232 | . . 3 ⊢ ((𝐴(,)𝐵) ⊆ (0(,)1) → (𝐴(,)𝐵) ∈ ((topGen‘ran (,)) ↾t (0[,]1))) |
14 | dfii2 24073 | . . 3 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
15 | 13, 14 | eleqtrrdi 2845 | . 2 ⊢ ((𝐴(,)𝐵) ⊆ (0(,)1) → (𝐴(,)𝐵) ∈ II) |
16 | 4, 15 | syl 17 | 1 ⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2101 Vcvv 3434 ⊆ wss 3889 class class class wbr 5077 ran crn 5592 ‘cfv 6447 (class class class)co 7295 0cc0 10899 1c1 10900 ℝ*cxr 11036 ≤ cle 11038 (,)cioo 13107 [,]cicc 13110 ↾t crest 17159 topGenctg 17176 Topctop 22070 IIcii 24066 |
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 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-sup 9229 df-inf 9230 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-z 12348 df-uz 12611 df-q 12717 df-rp 12759 df-xneg 12876 df-xadd 12877 df-xmul 12878 df-ioo 13111 df-icc 13114 df-seq 13750 df-exp 13811 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-rest 17161 df-topgen 17182 df-psmet 20617 df-xmet 20618 df-met 20619 df-bl 20620 df-mopn 20621 df-top 22071 df-topon 22088 df-bases 22124 df-ii 24068 |
This theorem is referenced by: (None) |
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