Theorem 0sal 41057

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 41051	. . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
2	1	ibi 256	. 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
3	2	simp1d 1136	1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ w3a 1071   ∈ wcel 2145  ∀wral 3061   ∖ cdif 3720  ∅c0 4063  𝒫 cpw 4297  ∪ cuni 4574   class class class wbr 4786  ωcom 7212   ≼ cdom 8107  SAlgcsalg 41045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-salg 41046

This theorem is referenced by:  saluni  41061  intsal  41065  0sald  41085  ismeannd  41201