Theorem unielsiga 33804

Description: A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)

Assertion

Ref	Expression
unielsiga	⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)

Proof of Theorem unielsiga

Step	Hyp	Ref	Expression
1		sgon 33800	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		baselsiga 33791	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → ∪ 𝑆 ∈ 𝑆)
3	1, 2	syl 17	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2098  ∪ cuni 4903  ran crn 5673  ‘cfv 6543  sigAlgebracsiga 33784

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-siga 33785

This theorem is referenced by:  mbfmcst  33936  1stmbfm  33937  2ndmbfm  33938  imambfm  33939  mbfmco  33941  br2base  33946  prob01  34090  probfinmeasb  34105