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Theorem sigasspw 32755
Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
Assertion
Ref Expression
sigasspw (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 βŠ† 𝒫 𝐴)

Proof of Theorem sigasspw
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elex 3466 . . 3 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 ∈ V)
2 issiga 32751 . . . 4 (𝑆 ∈ V β†’ (𝑆 ∈ (sigAlgebraβ€˜π΄) ↔ (𝑆 βŠ† 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))))
32biimpa 478 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebraβ€˜π΄)) β†’ (𝑆 βŠ† 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))))
41, 3mpancom 687 . 2 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ (𝑆 βŠ† 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))))
54simpld 496 1 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 βŠ† 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870   class class class wbr 5110  β€˜cfv 6501  Ο‰com 7807   β‰Ό 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-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-iota 6453  df-fun 6503  df-fv 6509  df-siga 32748
This theorem is referenced by:  elsigass  32764  insiga  32776  sigapisys  32794  sigaldsys  32798  brsigasspwrn  32824  1stmbfm  32900  2ndmbfm  32901
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