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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elsigass | Structured version Visualization version GIF version | ||
| Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.) |
| Ref | Expression |
|---|---|
| elsigass | β’ ((π β βͺ ran sigAlgebra β§ π΄ β π) β π΄ β βͺ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sgon 33784 | . . . 4 β’ (π β βͺ ran sigAlgebra β π β (sigAlgebraββͺ π)) | |
| 2 | sigasspw 33776 | . . . 4 β’ (π β (sigAlgebraββͺ π) β π β π« βͺ π) | |
| 3 | 1, 2 | syl 17 | . . 3 β’ (π β βͺ ran sigAlgebra β π β π« βͺ π) |
| 4 | 3 | sselda 3982 | . 2 β’ ((π β βͺ ran sigAlgebra β§ π΄ β π) β π΄ β π« βͺ π) |
| 5 | 4 | elpwid 4615 | 1 β’ ((π β βͺ ran sigAlgebra β§ π΄ β π) β π΄ β βͺ π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β wcel 2098 β wss 3949 π« cpw 4606 βͺ cuni 4912 ran crn 5683 βcfv 6553 sigAlgebracsiga 33768 |
| 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-ne 2938 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-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 df-siga 33769 |
| This theorem is referenced by: (None) |
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