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 41839 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) = (𝐴(,]𝐶)) |
4 | 3 | eqcomd 2829 | . 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 42645 | . . . . . 6 ⊢ 𝐵 ∈ SAlg |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐵 ∈ SAlg) |
9 | nnct 13352 | . . . . . 6 ⊢ ℕ ≼ ω | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≼ ω) |
11 | nnn0 41654 | . . . . . 6 ⊢ ℕ ≠ ∅ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≠ ∅) |
13 | 5, 6 | iooborel 42641 | . . . . . 6 ⊢ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵 |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
15 | 8, 10, 12, 14 | saliincl 42617 | . . . 4 ⊢ (⊤ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
16 | 15 | mptru 1544 | . . 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 2915 | 1 ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 ∩ ciin 4922 class class class wbr 5068 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ωcom 7582 ≼ cdom 8509 ℝcr 10538 1c1 10540 + caddc 10542 ℝ*cxr 10676 / cdiv 11299 ℕcn 11640 (,)cioo 12741 (,]cioc 12742 topGenctg 16713 SAlgcsalg 42600 SalGencsalgen 42604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-ioc 12746 df-fl 13165 df-topgen 16719 df-top 21504 df-bases 21556 df-salg 42601 df-salgen 42605 |
This theorem is referenced by: incsmflem 43025 decsmflem 43049 smfsuplem2 43093 |
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