Theorem 0sal 46316

Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0sal	⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Proof of Theorem 0sal

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issal 46310	. . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
2	1	ibi 267	. 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
3	2	simp1d 1142	1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3052   ∖ cdif 3928  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888   class class class wbr 5124  ωcom 7866   ≼ cdom 8962  SAlgcsalg 46304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-ss 3948  df-pw 4582  df-uni 4889  df-salg 46305

This theorem is referenced by:  saluni  46321  intsal  46326  0sald  46346  ismeannd  46463