Metamath Proof Explorer

Theorem shjcl

Description: Closure of join in SH . (Contributed by NM, 2-Nov-1999) (New usage is discouraged.)

Ref Expression
Assertion shjcl 
|- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) e. CH )

Proof

Step Hyp Ref Expression
1 shss 
 |-  ( A e. SH -> A C_ ~H )
2 shss 
 |-  ( B e. SH -> B C_ ~H )
3 sshjcl 
 |-  ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) e. CH )
4 1 2 3 syl2an 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A vH B ) e. CH )