Metamath Proof Explorer


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 )