Theorem 0sald 46705

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 46675	. 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2114  ∅c0 4287  SAlgcsalg 46663

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-ss 3920  df-pw 4558  df-uni 4866  df-salg 46664

This theorem is referenced by:  subsalsal  46714  smfpimltxr  47102  smfconst  47104  smfpimgtxr  47135  smfresal  47143