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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 32763 | . 2 β’ (π β βͺ ran sigAlgebra β π β (sigAlgebraββͺ π)) | |
2 | elex 3466 | . . 3 β’ (π β βͺ ran sigAlgebra β π β V) | |
3 | issiga 32751 | . . 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 397 β§ w3a 1088 β wcel 2107 βwral 3065 Vcvv 3448 β cdif 3912 β wss 3915 π« cpw 4565 βͺ cuni 4870 class class class wbr 5110 ran crn 5639 βcfv 6501 Οcom 7807 βΌ cdom 8888 sigAlgebracsiga 32747 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 df-siga 32748 |
This theorem is referenced by: sigaclci 32771 difelsiga 32772 unelsiga 32773 cntmeas 32865 probfinmeasbALTV 33069 |
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