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Mirrors > Home > MPE Home > Th. List > Mathboxes > brsigarn | Structured version Visualization version GIF version |
Description: The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.) |
Ref | Expression |
---|---|
brsigarn | ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . 3 ⊢ (topGen‘ran (,)) ∈ V | |
2 | sigagensiga 31510 | . . 3 ⊢ ((topGen‘ran (,)) ∈ V → (sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,)))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,))) |
4 | df-brsiga 31551 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | uniretop 23368 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
6 | 5 | fveq2i 6648 | . 2 ⊢ (sigAlgebra‘ℝ) = (sigAlgebra‘∪ (topGen‘ran (,))) |
7 | 3, 4, 6 | 3eltr4i 2903 | 1 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 ℝcr 10525 (,)cioo 12726 topGenctg 16703 sigAlgebracsiga 31477 sigaGencsigagen 31507 𝔅ℝcbrsiga 31550 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 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-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-topgen 16709 df-bases 21551 df-siga 31478 df-sigagen 31508 df-brsiga 31551 |
This theorem is referenced by: brsigasspwrn 31554 mbfmvolf 31634 elmbfmvol2 31635 mbfmcnt 31636 br2base 31637 dya2iocbrsiga 31643 dya2icobrsiga 31644 sxbrsigalem5 31656 sxbrsiga 31658 isrrvv 31811 rrvadd 31820 rrvmulc 31821 dstrvprob 31839 |
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