Theorem baselsiga 34135

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 3457	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2		issiga 34132	. . . 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 499	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
4	3	simp1d 1142	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴 ∈ 𝑆)
5	1, 4	mpancom 688	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897  𝒫 cpw 4549  ∪ cuni 4858   class class class wbr 5093  ‘cfv 6487  ωcom 7802   ≼ cdom 8873  sigAlgebracsiga 34128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fv 6495  df-siga 34129

This theorem is referenced by:  unielsiga  34148  sigaldsys  34179  cldssbrsiga  34207  1stmbfm  34280  2ndmbfm  34281  unveldomd  34435  probmeasb  34450  dstrvprob  34492