Theorem baselsiga 30487

Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)

Assertion

Ref	Expression
baselsiga	⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆)

Proof of Theorem baselsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3352	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2		issiga 30483	. . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
3	2	simplbda 655	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
4	3	simp1d 1137	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴 ∈ 𝑆)
5	1, 4	mpancom 706	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ∖ cdif 3712   ⊆ wss 3715  𝒫 cpw 4302  ∪ cuni 4588   class class class wbr 4804  ‘cfv 6049  ωcom 7230   ≼ cdom 8119  sigAlgebracsiga 30479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-siga 30480

This theorem is referenced by:  unielsiga  30500  sigaldsys  30531  cldssbrsiga  30559  1stmbfm  30631  2ndmbfm  30632  unveldomd  30786  probmeasb  30801  dstrvprob  30842