Theorem chlubii

Description: Hilbert lattice join is the least upper bound of two elements (one direction of chlubi ). (Contributed by NM, 15-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	chlubii	|- ( ( A C_ C /\ B C_ C ) -> ( A vH B ) C_ 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 2 3	chlubi	|- ( ( A C_ C /\ B C_ C ) <-> ( A vH B ) C_ C )
5	4	biimpi	|- ( ( A C_ C /\ B C_ C ) -> ( A vH B ) C_ C )