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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 6919 | . . 3 ⊢ (topGen‘ran (,)) ∈ V | |
| 2 | sigagensiga 34142 | . . 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 34183 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
| 5 | uniretop 24783 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 6 | 5 | fveq2i 6909 | . 2 ⊢ (sigAlgebra‘ℝ) = (sigAlgebra‘∪ (topGen‘ran (,))) |
| 7 | 3, 4, 6 | 3eltr4i 2854 | 1 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 ℝcr 11154 (,)cioo 13387 topGenctg 17482 sigAlgebracsiga 34109 sigaGencsigagen 34139 𝔅ℝcbrsiga 34182 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-bases 22953 df-siga 34110 df-sigagen 34140 df-brsiga 34183 |
| This theorem is referenced by: brsigasspwrn 34186 mbfmvolf 34268 elmbfmvol2 34269 mbfmcnt 34270 br2base 34271 dya2iocbrsiga 34277 dya2icobrsiga 34278 sxbrsigalem5 34290 sxbrsiga 34292 isrrvv 34445 rrvadd 34454 rrvmulc 34455 dstrvprob 34474 |
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