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Theorem baselsiga 31381
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
Assertion
Ref Expression
baselsiga (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)

Proof of Theorem baselsiga
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3489 . 2 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2 issiga 31378 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simplbda 503 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
43simp1d 1139 . 2 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴𝑆)
51, 4mpancom 687 1 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wral 3126  Vcvv 3471  cdif 3907  wss 3910  𝒫 cpw 4512   cuni 4811   class class class wbr 5039  cfv 6328  ωcom 7555  cdom 8482  sigAlgebracsiga 31374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-siga 31375
This theorem is referenced by:  unielsiga  31394  sigaldsys  31425  cldssbrsiga  31453  1stmbfm  31525  2ndmbfm  31526  unveldomd  31680  probmeasb  31695  dstrvprob  31736
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