Theorem 0sald 42990

Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
0sald.1	⊢ (𝜑 → 𝑆 ∈ SAlg)

Assertion

Ref	Expression
0sald	⊢ (𝜑 → ∅ ∈ 𝑆)

Proof of Theorem 0sald

Step	Hyp	Ref	Expression
1		0sald.1	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2		0sal 42962	. 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2111  ∅c0 4243  SAlgcsalg 42950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-salg 42951

This theorem is referenced by:  subsalsal  42999  smfpimltxr  43381  smfconst  43383  smfpimgtxr  43413  smfresal  43420