Theorem 0sald 40568

Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
0sald.1	|- ( ph -> S e. SAlg )

Assertion

Ref	Expression
0sald	|- ( ph -> (/) e. S )

Proof of Theorem 0sald

Step	Hyp	Ref	Expression
1		0sald.1	. 2 |- ( ph -> S e. SAlg )
2		0sal 40540	. 2 |- ( S e. SAlg -> (/) e. S )
3	1, 2	syl 17	1 |- ( ph -> (/) e. S )

Syntax hints:   -> wi 4   e. wcel 1990   (/)c0 3915  SAlgcsalg 40528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-salg 40529

This theorem is referenced by:  subsalsal  40577  smfpimltxr  40956  smfconst  40958  smfpimgtxr  40988  smfresal  40995