Metamath Proof Explorer


Theorem chlej1i

Description: Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1
|- A e. CH
chjcl.2
|- B e. CH
chlub.1
|- C e. CH
Assertion chlej1i
|- ( A C_ B -> ( A vH C ) C_ ( B vH C ) )

Proof

Step Hyp Ref Expression
1 ch0le.1
 |-  A e. CH
2 chjcl.2
 |-  B e. CH
3 chlub.1
 |-  C e. CH
4 1 chshii
 |-  A e. SH
5 2 chshii
 |-  B e. SH
6 3 chshii
 |-  C e. SH
7 4 5 6 shlej1i
 |-  ( A C_ B -> ( A vH C ) C_ ( B vH C ) )