Theorem 0sal 46340

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 46334	. . 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 2107  ∀wral 3060   ∖ cdif 3947  ∅c0 4332  𝒫 cpw 4599  ∪ cuni 4906   class class class wbr 5142  ωcom 7888   ≼ cdom 8984  SAlgcsalg 46328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-ss 3967  df-pw 4601  df-uni 4907  df-salg 46329

This theorem is referenced by:  saluni  46345  intsal  46350  0sald  46370  ismeannd  46487