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Theorem baselsiga 30780
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 3414 . 2 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2 issiga 30776 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simplbda 495 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
43simp1d 1133 . 2 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴𝑆)
51, 4mpancom 678 1 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071  wcel 2107  wral 3090  Vcvv 3398  cdif 3789  wss 3792  𝒫 cpw 4379   cuni 4673   class class class wbr 4888  cfv 6137  ωcom 7345  cdom 8241  sigAlgebracsiga 30772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-siga 30773
This theorem is referenced by:  unielsiga  30793  sigaldsys  30824  cldssbrsiga  30852  1stmbfm  30924  2ndmbfm  30925  unveldomd  31080  probmeasb  31095  dstrvprob  31136
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