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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 24071 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | unisg 32570 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ (sigaGen‘(topGen‘ran (,))) = ∪ (topGen‘ran (,)) |
4 | df-brsiga 32609 | . . 3 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
5 | 4 | unieqi 4876 | . 2 ⊢ ∪ 𝔅ℝ = ∪ (sigaGen‘(topGen‘ran (,))) |
6 | uniretop 24072 | . 2 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
7 | 3, 5, 6 | 3eqtr4i 2774 | 1 ⊢ ∪ 𝔅ℝ = ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∪ cuni 4863 ran crn 5632 ‘cfv 6493 ℝcr 11008 (,)cioo 13218 topGenctg 17273 Topctop 22188 sigaGencsigagen 32565 𝔅ℝcbrsiga 32608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-ioo 13222 df-topgen 17279 df-top 22189 df-bases 22242 df-siga 32536 df-sigagen 32566 df-brsiga 32609 |
This theorem is referenced by: elmbfmvol2 32695 mbfmcnt 32696 br2base 32697 isrrvv 32871 orvcelval 32896 dstrvprob 32899 |
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