Theorem infcl

Description: An infimum belongs to its base class (closure law). See also inflb and infglb . (Contributed by AV, 3-Sep-2020)

	Ref	Expression
Hypotheses	infcl.1	|- ( ph -> R Or A )
	infcl.2	|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
Assertion	infcl	|- ( ph -> inf ( B , A , R ) e. A )

Proof

Step	Hyp	Ref	Expression
1		infcl.1	|- ( ph -> R Or A )
2		infcl.2	|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
3		df-inf	|- inf ( B , A , R ) = sup ( B , A , `' R )
4		cnvso	|- ( R Or A <-> `' R Or A )
5	1 4	sylib	|- ( ph -> `' R Or A )
6	1 2	infcllem	|- ( ph -> E. x e. A ( A. y e. B -. x `' R y /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) )
7	5 6	supcl	|- ( ph -> sup ( B , A , `' R ) e. A )
8	3 7	eqeltrid	|- ( ph -> inf ( B , A , R ) e. A )