Theorem 0sald 46370

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 46340	. 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2107  ∅c0 4332  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:  subsalsal  46379  smfpimltxr  46767  smfconst  46769  smfpimgtxr  46800  smfresal  46808