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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iocborel | Structured version Visualization version GIF version |
Description: A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
iocborel.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
iocborel.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
iocborel.t | ⊢ 𝐽 = (topGen‘ran (,)) |
iocborel.b | ⊢ 𝐵 = (SalGen‘𝐽) |
Ref | Expression |
---|---|
iocborel | ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocborel.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | iocborel.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
3 | 1, 2 | iooiinioc 40705 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) = (𝐴(,]𝐶)) |
4 | 3 | eqcomd 2784 | . 2 ⊢ (𝜑 → (𝐴(,]𝐶) = ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛)))) |
5 | iocborel.t | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
6 | iocborel.b | . . . . . . 7 ⊢ 𝐵 = (SalGen‘𝐽) | |
7 | 5, 6 | bor1sal 41511 | . . . . . 6 ⊢ 𝐵 ∈ SAlg |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐵 ∈ SAlg) |
9 | nnct 13104 | . . . . . 6 ⊢ ℕ ≼ ω | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≼ ω) |
11 | nnn0 40517 | . . . . . 6 ⊢ ℕ ≠ ∅ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≠ ∅) |
13 | 5, 6 | iooborel 41507 | . . . . . 6 ⊢ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵 |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
15 | 8, 10, 12, 14 | saliincl 41483 | . . . 4 ⊢ (⊤ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
16 | 15 | mptru 1609 | . . 3 ⊢ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵 |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
18 | 4, 17 | eqeltrd 2859 | 1 ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ⊤wtru 1602 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 ∩ ciin 4756 class class class wbr 4888 ran crn 5358 ‘cfv 6137 (class class class)co 6924 ωcom 7345 ≼ cdom 8241 ℝcr 10273 1c1 10275 + caddc 10277 ℝ*cxr 10412 / cdiv 11035 ℕcn 11379 (,)cioo 12492 (,]cioc 12493 topGenctg 16495 SAlgcsalg 41466 SalGencsalgen 41470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-card 9100 df-acn 9103 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-q 12101 df-rp 12143 df-ioo 12496 df-ioc 12497 df-fl 12917 df-topgen 16501 df-top 21117 df-bases 21169 df-salg 41467 df-salgen 41471 |
This theorem is referenced by: incsmflem 41891 decsmflem 41915 smfsuplem2 41959 |
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