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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 32587 | . 2 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
2 | sigasspw 32519 | . 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 2107 ⊆ wss 3909 𝒫 cpw 4559 ‘cfv 6494 ℝcr 11009 sigAlgebracsiga 32511 𝔅ℝcbrsiga 32584 |
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 7665 ax-cnex 11066 ax-resscn 11067 ax-pre-lttri 11084 ax-pre-lttrn 11085 |
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 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7914 df-2nd 7915 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-ioo 13223 df-topgen 17285 df-bases 22248 df-siga 32512 df-sigagen 32542 df-brsiga 32585 |
This theorem is referenced by: br2base 32673 sxbrsigalem2 32690 sxbrsiga 32694 |
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