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Theorem baselsiga 29956
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 3198 . 2 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2 issiga 29952 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simplbda 653 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
43simp1d 1071 . 2 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴𝑆)
51, 4mpancom 702 1 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cfv 5847  ωcom 7012  cdom 7897  sigAlgebracsiga 29948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-siga 29949
This theorem is referenced by:  unielsiga  29969  sigaldsys  30000  cldssbrsiga  30028  1stmbfm  30100  2ndmbfm  30101  unveldomd  30255  probmeasb  30270  dstrvprob  30311
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