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Theorem baselsiga 33791
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
Assertion
Ref Expression
baselsiga (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝐴 ∈ 𝑆)

Proof of Theorem baselsiga
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . 2 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝑆 ∈ V)
2 issiga 33788 . . . 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)))))
32simplbda 498 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebraβ€˜π΄)) β†’ (𝐴 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))
43simp1d 1139 . 2 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebraβ€˜π΄)) β†’ 𝐴 ∈ 𝑆)
51, 4mpancom 686 1 (𝑆 ∈ (sigAlgebraβ€˜π΄) β†’ 𝐴 ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463   βˆ– cdif 3936   βŠ† wss 3939  π’« cpw 4598  βˆͺ cuni 4903   class class class wbr 5143  β€˜cfv 6543  Ο‰com 7868   β‰Ό cdom 8960  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-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-iota 6495  df-fun 6545  df-fv 6551  df-siga 33785
This theorem is referenced by:  unielsiga  33804  sigaldsys  33835  cldssbrsiga  33863  1stmbfm  33937  2ndmbfm  33938  unveldomd  34092  probmeasb  34107  dstrvprob  34148
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