Theorem saluni 40544

Description: A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
saluni	|- ( S e. SAlg -> U. S e. S )

Proof of Theorem saluni

Step	Hyp	Ref	Expression
1		dif0 3950	. 2 |- ( U. S \ (/) ) = U. S
2		0sal 40540	. . 3 |- ( S e. SAlg -> (/) e. S )
3		saldifcl 40539	. . 3 |- ( ( S e. SAlg /\ (/) e. S ) -> ( U. S \ (/) ) e. S )
4	2, 3	mpdan 702	. 2 |- ( S e. SAlg -> ( U. S \ (/) ) e. S )
5	1, 4	syl5eqelr 2706	1 |- ( S e. SAlg -> U. S e. S )

Syntax hints:   -> wi 4   e. wcel 1990   \ cdif 3571   (/)c0 3915   U.cuni 4436  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-nul 3916  df-pw 4160  df-uni 4437  df-salg 40529

This theorem is referenced by:  intsaluni  40547  unisalgen  40558  salgencntex  40561  salunid  40571