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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 45509 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐶 + (1 / 𝑛))) = (𝐴(,]𝐶)) |
4 | 3 | eqcomd 2741 | . 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 46311 | . . . . . 6 ⊢ 𝐵 ∈ SAlg |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐵 ∈ SAlg) |
9 | nnct 14019 | . . . . . 6 ⊢ ℕ ≼ ω | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≼ ω) |
11 | nnn0 45328 | . . . . . 6 ⊢ ℕ ≠ ∅ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℕ ≠ ∅) |
13 | 5, 6 | iooborel 46307 | . . . . . 6 ⊢ (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵 |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐴(,)(𝐶 + (1 / 𝑛))) ∈ 𝐵) |
15 | 8, 10, 12, 14 | saliincl 46283 | . . . 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 2839 | 1 ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ∩ ciin 4997 class class class wbr 5148 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 ℝcr 11152 1c1 11154 + caddc 11156 ℝ*cxr 11292 / cdiv 11918 ℕcn 12264 (,)cioo 13384 (,]cioc 13385 topGenctg 17484 SAlgcsalg 46264 SalGencsalgen 46268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-ioo 13388 df-ioc 13389 df-fl 13829 df-topgen 17490 df-top 22916 df-bases 22969 df-salg 46265 df-salgen 46269 |
This theorem is referenced by: incsmflem 46697 decsmflem 46722 smfsuplem2 46768 |
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