Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > isrnsigau | Structured version Visualization version GIF version | ||
| Description: The property of being a sigma-algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.) |
| Ref | Expression |
|---|---|
| isrnsigau | β’ (π β βͺ ran sigAlgebra β (π β π« βͺ π β§ (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sgon 33652 | . 2 β’ (π β βͺ ran sigAlgebra β π β (sigAlgebraββͺ π)) | |
| 2 | elex 3487 | . . 3 β’ (π β βͺ ran sigAlgebra β π β V) | |
| 3 | issiga 33640 | . . 3 β’ (π β V β (π β (sigAlgebraββͺ π) β (π β π« βͺ π β§ (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))))) | |
| 4 | 2, 3 | syl 17 | . 2 β’ (π β βͺ ran sigAlgebra β (π β (sigAlgebraββͺ π) β (π β π« βͺ π β§ (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))))) |
| 5 | 1, 4 | mpbid 231 | 1 β’ (π β βͺ ran sigAlgebra β (π β π« βͺ π β§ (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 β wcel 2098 βwral 3055 Vcvv 3468 β cdif 3940 β wss 3943 π« cpw 4597 βͺ cuni 4902 class class class wbr 5141 ran crn 5670 βcfv 6537 Οcom 7852 βΌ cdom 8939 sigAlgebracsiga 33636 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-fv 6545 df-siga 33637 |
| This theorem is referenced by: sigaclci 33660 difelsiga 33661 unelsiga 33662 cntmeas 33754 probfinmeasbALTV 33958 |
| Copyright terms: Public domain | W3C validator |