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Mathbox for Thierry Arnoux |
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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 24148 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | unisg 32806 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,)) |
4 | df-brsiga 32845 | . . 3 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | 4 | unieqi 4882 | . 2 ⊢ ∪ 𝔅ℝ = ∪ (sigaGen‘(topGen‘ran (,))) |
6 | uniretop 24149 | . 2 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
7 | 3, 5, 6 | 3eqtr4i 2771 | 1 ⊢ ∪ 𝔅ℝ = ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∪ cuni 4869 ran crn 5638 ‘cfv 6500 ℝcr 11058 (,)cioo 13273 topGenctg 17327 Topctop 22265 sigaGencsigagen 32801 𝔅ℝcbrsiga 32844 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-ioo 13277 df-topgen 17333 df-top 22266 df-bases 22319 df-siga 32772 df-sigagen 32802 df-brsiga 32845 |
This theorem is referenced by: elmbfmvol2 32931 mbfmcnt 32932 br2base 32933 isrrvv 33107 orvcelval 33132 dstrvprob 33135 |
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