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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 6920 | . . 3 ⊢ (topGen‘ran (,)) ∈ V | |
2 | sigagensiga 34122 | . . 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 34163 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | uniretop 24799 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
6 | 5 | fveq2i 6910 | . 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 2106 Vcvv 3478 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 ℝcr 11152 (,)cioo 13384 topGenctg 17484 sigAlgebracsiga 34089 sigaGencsigagen 34119 𝔅ℝcbrsiga 34162 |
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-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
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-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 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-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 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-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 df-topgen 17490 df-bases 22969 df-siga 34090 df-sigagen 34120 df-brsiga 34163 |
This theorem is referenced by: brsigasspwrn 34166 mbfmvolf 34248 elmbfmvol2 34249 mbfmcnt 34250 br2base 34251 dya2iocbrsiga 34257 dya2icobrsiga 34258 sxbrsigalem5 34270 sxbrsiga 34272 isrrvv 34425 rrvadd 34434 rrvmulc 34435 dstrvprob 34453 |
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