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Theorem sigasspw 33644
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 3487 . . 3 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 ∈ V)
2 issiga 33640 . . . 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 476 . . 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 685 . 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 494 1 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 βŠ† 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βˆ– cdif 3940   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902   class class class wbr 5141  β€˜cfv 6537  Ο‰com 7852   β‰Ό cdom 8939  sigAlgebracsiga 33636
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-siga 33637
This theorem is referenced by:  elsigass  33653  insiga  33665  sigapisys  33683  sigaldsys  33687  brsigasspwrn  33713  1stmbfm  33789  2ndmbfm  33790
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