Theorem chlubi

Description: Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004) (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	chlubi	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C$$

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	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
5	2	chshii	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
6	4 5 3	shlubi	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C$$