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Metamath Proof Explorer

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	$⊢ A \in C_{ℋ}$
	chjcl.2	$⊢ B \in C_{ℋ}$
	chlub.1	$⊢ C \in C_{ℋ}$
Assertion	chlubi	$⊢ (A \subseteq C \land B \subseteq C) \leftrightarrow A \lor_{ℋ} B \subseteq C$

Step	Hyp	Ref	Expression
1		ch0le.1	$⊢ A \in C_{ℋ}$
2		chjcl.2	$⊢ B \in C_{ℋ}$
3		chlub.1	$⊢ C \in C_{ℋ}$
4	1	chshii	$⊢ A \in S_{ℋ}$
5	2	chshii	$⊢ B \in S_{ℋ}$
6	4 5 3	shlubi	$⊢ (A \subseteq C \land B \subseteq C) \leftrightarrow A \lor_{ℋ} B \subseteq C$