Theorem difunielsiga

Description: A sigma-algebra is closed under complement relative to its base set. This is immediate from the definition, see issiga , but the library states it nowhere in this form. (Contributed by Vincent Gonzalez, 17-Aug-2026)

	Ref	Expression
Assertion	difunielsiga	|- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( U. S \ A ) e. S )

Proof

Step	Hyp	Ref	Expression
1		isrnsigau	|- ( S e. U. ran sigAlgebra -> ( S C_ ~P U. S /\ ( U. S e. S /\ A. x e. S ( U. S \ x ) e. S /\ A. x e. ~P S ( x ~<_ _om -> U. x e. S ) ) ) )
2	1	simprd	|- ( S e. U. ran sigAlgebra -> ( U. S e. S /\ A. x e. S ( U. S \ x ) e. S /\ A. x e. ~P S ( x ~<_ _om -> U. x e. S ) ) )
3	2	simp2d	|- ( S e. U. ran sigAlgebra -> A. x e. S ( U. S \ x ) e. S )
4		difeq2	|- ( x = A -> ( U. S \ x ) = ( U. S \ A ) )
5	4	eleq1d	|- ( x = A -> ( ( U. S \ x ) e. S <-> ( U. S \ A ) e. S ) )
6	5	rspccva	|- ( ( A. x e. S ( U. S \ x ) e. S /\ A e. S ) -> ( U. S \ A ) e. S )
7	3 6	sylan	|- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( U. S \ A ) e. S )