Theorem 0sal 47054

Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0sal	⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Proof of Theorem 0sal

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issal 47048	. . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
2	1	ibi 270	. 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
3	2	simp1d 1160	1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562  ∪ cuni 4872   class class class wbr 5109  ωcom 7858   ≼ cdom 8937  SAlgcsalg 47042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-ss 3922  df-pw 4564  df-uni 4873  df-salg 47043

This theorem is referenced by:  saluni  47059  intsal  47064  0sald  47084  ismeannd  47201