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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigasspw | Structured version Visualization version GIF version |
Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.) |
Ref | Expression |
---|---|
sigasspw | β’ (π β (sigAlgebraβπ΄) β π β π« π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3492 | . . 3 β’ (π β (sigAlgebraβπ΄) β π β V) | |
2 | issiga 33764 | . . . 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 | biimpa 475 | . . 3 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)))) |
4 | 1, 3 | mpancom 686 | . 2 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)))) |
5 | 4 | simpld 493 | 1 β’ (π β (sigAlgebraβπ΄) β π β π« π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 β wcel 2098 βwral 3058 Vcvv 3473 β cdif 3946 β wss 3949 π« cpw 4606 βͺ cuni 4912 class class class wbr 5152 βcfv 6553 Οcom 7876 βΌ cdom 8968 sigAlgebracsiga 33760 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-siga 33761 |
This theorem is referenced by: elsigass 33777 insiga 33789 sigapisys 33807 sigaldsys 33811 brsigasspwrn 33837 1stmbfm 33913 2ndmbfm 33914 |
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