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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 42193 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) = (𝐴(,]𝐶)) |
4 | 3 | eqcomd 2804 | . 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 42995 | . . . . . 6 ⊢ 𝐵 ∈ SAlg |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐵 ∈ SAlg) |
9 | nnct 13344 | . . . . . 6 ⊢ ℕ ≼ ω | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≼ ω) |
11 | nnn0 42011 | . . . . . 6 ⊢ ℕ ≠ ∅ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≠ ∅) |
13 | 5, 6 | iooborel 42991 | . . . . . 6 ⊢ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵 |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
15 | 8, 10, 12, 14 | saliincl 42967 | . . . 4 ⊢ (⊤ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
16 | 15 | mptru 1545 | . . 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 2890 | 1 ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ∩ ciin 4882 class class class wbr 5030 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ωcom 7560 ≼ cdom 8490 ℝcr 10525 1c1 10527 + caddc 10529 ℝ*cxr 10663 / cdiv 11286 ℕcn 11625 (,)cioo 12726 (,]cioc 12727 topGenctg 16703 SAlgcsalg 42950 SalGencsalgen 42954 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ioc 12731 df-fl 13157 df-topgen 16709 df-top 21499 df-bases 21551 df-salg 42951 df-salgen 42955 |
This theorem is referenced by: incsmflem 43375 decsmflem 43399 smfsuplem2 43443 |
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