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Mirrors > Home > MPE Home > Th. List > Mathboxes > unibrsiga | Structured version Visualization version GIF version |
Description: The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
unibrsiga | ⊢ ∪ 𝔅ℝ = ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 22565 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | unisg 30206 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,)) |
4 | df-brsiga 30245 | . . 3 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | 4 | unieqi 4445 | . 2 ⊢ ∪ 𝔅ℝ = ∪ (sigaGen‘(topGen‘ran (,))) |
6 | uniretop 22566 | . 2 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
7 | 3, 5, 6 | 3eqtr4i 2654 | 1 ⊢ ∪ 𝔅ℝ = ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 ran crn 5115 ‘cfv 5888 ℝcr 9935 (,)cioo 12175 topGenctg 16098 Topctop 20698 sigaGencsigagen 30201 𝔅ℝcbrsiga 30244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-topgen 16104 df-top 20699 df-bases 20750 df-siga 30171 df-sigagen 30202 df-brsiga 30245 |
This theorem is referenced by: elmbfmvol2 30329 mbfmcnt 30330 br2base 30331 isrrvv 30505 orvcelval 30530 dstrvprob 30533 |
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