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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 6851 | . . 3 ⊢ (topGen‘ran (,)) ∈ V | |
2 | sigagensiga 32520 | . . 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 32561 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | uniretop 24054 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
6 | 5 | fveq2i 6841 | . 2 ⊢ (sigAlgebra‘ℝ) = (sigAlgebra‘∪ (topGen‘ran (,))) |
7 | 3, 4, 6 | 3eltr4i 2852 | 1 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3444 ∪ cuni 4864 ran crn 5632 ‘cfv 6492 ℝcr 10984 (,)cioo 13194 topGenctg 17255 sigAlgebracsiga 32487 sigaGencsigagen 32517 𝔅ℝcbrsiga 32560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-pre-lttri 11059 ax-pre-lttrn 11060 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-1st 7912 df-2nd 7913 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-ioo 13198 df-topgen 17261 df-bases 22224 df-siga 32488 df-sigagen 32518 df-brsiga 32561 |
This theorem is referenced by: brsigasspwrn 32564 mbfmvolf 32646 elmbfmvol2 32647 mbfmcnt 32648 br2base 32649 dya2iocbrsiga 32655 dya2icobrsiga 32656 sxbrsigalem5 32668 sxbrsiga 32670 isrrvv 32823 rrvadd 32832 rrvmulc 32833 dstrvprob 32851 |
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