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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 6113 | . . 3 ⊢ (topGen‘ran (,)) ∈ V | |
2 | sigagensiga 29531 | . . 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 29572 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | uniretop 22376 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
6 | 5 | fveq2i 6106 | . 2 ⊢ (sigAlgebra‘ℝ) = (sigAlgebra‘∪ (topGen‘ran (,))) |
7 | 3, 4, 6 | 3eltr4i 2701 | 1 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 ℝcr 9814 (,)cioo 12046 topGenctg 15921 sigAlgebracsiga 29497 sigaGencsigagen 29528 𝔅ℝcbrsiga 29571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-topgen 15927 df-bases 20522 df-siga 29498 df-sigagen 29529 df-brsiga 29572 |
This theorem is referenced by: brsigasspwrn 29575 mbfmvolf 29655 elmbfmvol2 29656 mbfmcnt 29657 br2base 29658 dya2iocbrsiga 29664 dya2icobrsiga 29665 sxbrsigalem5 29677 sxbrsiga 29679 isrrvv 29832 rrvadd 29841 rrvmulc 29842 dstrvprob 29860 |
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