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	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
	chlub.1	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chlubii	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\to A{\vee }_{\mathscr{H}}B\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chlub.1	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
4	1 2 3	chlubi	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C$$
5	4	biimpi	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\to A{\vee }_{\mathscr{H}}B\subseteq C$$