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Mathbox for Thierry Arnoux |
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| 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 33800 | . 2 β’ (π β βͺ ran sigAlgebra β π β (sigAlgebraββͺ π)) | |
| 2 | elex 3482 | . . 3 β’ (π β βͺ ran sigAlgebra β π β V) | |
| 3 | issiga 33788 | . . 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 394 β§ w3a 1084 β wcel 2098 βwral 3051 Vcvv 3463 β cdif 3936 β wss 3939 π« cpw 4598 βͺ cuni 4903 class class class wbr 5143 ran crn 5673 βcfv 6543 Οcom 7868 βΌ cdom 8960 sigAlgebracsiga 33784 |
| 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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-siga 33785 |
| This theorem is referenced by: sigaclci 33808 difelsiga 33809 unelsiga 33810 cntmeas 33902 probfinmeasbALTV 34106 |
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