Theorem 0sal 45336

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 45330	. . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
2	1	ibi 266	. 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
3	2	simp1d 1140	1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2104  ∀wral 3059   ∖ cdif 3946  ∅c0 4323  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  ωcom 7859   ≼ cdom 8941  SAlgcsalg 45324

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-salg 45325

This theorem is referenced by:  saluni  45341  intsal  45346  0sald  45366  ismeannd  45483