![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > baselsiga | Structured version Visualization version GIF version |
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.) |
Ref | Expression |
---|---|
baselsiga | β’ (π β (sigAlgebraβπ΄) β π΄ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3482 | . 2 β’ (π β (sigAlgebraβπ΄) β π β V) | |
2 | issiga 33788 | . . . 4 β’ (π β V β (π β (sigAlgebraβπ΄) β (π β π« π΄ β§ (π΄ β π β§ βπ₯ β π (π΄ β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))))) | |
3 | 2 | simplbda 498 | . . 3 β’ ((π β V β§ π β (sigAlgebraβπ΄)) β (π΄ β π β§ βπ₯ β π (π΄ β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))) |
4 | 3 | simp1d 1139 | . 2 β’ ((π β V β§ π β (sigAlgebraβπ΄)) β π΄ β π) |
5 | 1, 4 | mpancom 686 | 1 β’ (π β (sigAlgebraβπ΄) β π΄ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 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 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-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-iota 6495 df-fun 6545 df-fv 6551 df-siga 33785 |
This theorem is referenced by: unielsiga 33804 sigaldsys 33835 cldssbrsiga 33863 1stmbfm 33937 2ndmbfm 33938 unveldomd 34092 probmeasb 34107 dstrvprob 34148 |
Copyright terms: Public domain | W3C validator |