Theorem 0sald 46346

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 46316	. 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2109  ∅c0 4313  SAlgcsalg 46304

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-ss 3948  df-pw 4582  df-uni 4889  df-salg 46305

This theorem is referenced by:  subsalsal  46355  smfpimltxr  46743  smfconst  46745  smfpimgtxr  46776  smfresal  46784