![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigagenid | Structured version Visualization version GIF version |
Description: The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
sigagenid | β’ (π β βͺ ran sigAlgebra β (sigaGenβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgon 32787 | . . 3 β’ (π β βͺ ran sigAlgebra β π β (sigAlgebraββͺ π)) | |
2 | ssid 3970 | . . 3 β’ π β π | |
3 | sigagenss 32812 | . . 3 β’ ((π β (sigAlgebraββͺ π) β§ π β π) β (sigaGenβπ) β π) | |
4 | 1, 2, 3 | sylancl 587 | . 2 β’ (π β βͺ ran sigAlgebra β (sigaGenβπ) β π) |
5 | sssigagen 32808 | . 2 β’ (π β βͺ ran sigAlgebra β π β (sigaGenβπ)) | |
6 | 4, 5 | eqssd 3965 | 1 β’ (π β βͺ ran sigAlgebra β (sigaGenβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3914 βͺ cuni 4869 ran crn 5638 βcfv 6500 sigAlgebracsiga 32771 sigaGencsigagen 32801 |
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 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 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-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 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-fv 6508 df-siga 32772 df-sigagen 32802 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |