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Mirrors > Home > MPE Home > Th. List > Mathboxes > brsigasspwrn | Structured version Visualization version GIF version |
Description: The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.) |
Ref | Expression |
---|---|
brsigasspwrn | ⊢ 𝔅ℝ ⊆ 𝒫 ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsigarn 31888 | . 2 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
2 | sigasspw 31820 | . 2 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ⊆ 𝒫 ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4528 ‘cfv 6398 ℝcr 10753 sigAlgebracsiga 31812 𝔅ℝcbrsiga 31885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-pre-lttri 10828 ax-pre-lttrn 10829 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-po 5483 df-so 5484 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-1st 7780 df-2nd 7781 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-ioo 12964 df-topgen 16973 df-bases 21867 df-siga 31813 df-sigagen 31843 df-brsiga 31886 |
This theorem is referenced by: br2base 31972 sxbrsigalem2 31989 sxbrsiga 31993 |
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