Theorem 0sald 46923

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 46893	. 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3	1, 2	syl 18	1 ⊢ (𝜑 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2145  ∅c0 4288  SAlgcsalg 46881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-ss 3924  df-pw 4560  df-uni 4868  df-salg 46882

This theorem is referenced by:  subsalsal  46932  smfpimltxr  47320  smfconst  47322  smfpimgtxr  47353  smfresal  47361