This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem chjcl

Description: Closure of join in CH . (Contributed by NM, 2-Nov-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	chjcl	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \lor_{ℋ} B \in C_{ℋ}$

Step	Hyp	Ref	Expression
1		chsh	$⊢ A \in C_{ℋ} \to A \in S_{ℋ}$
2		chsh	$⊢ B \in C_{ℋ} \to B \in S_{ℋ}$
3		shjcl	$⊢ (A \in S_{ℋ} \land B \in S_{ℋ}) \to A \lor_{ℋ} B \in C_{ℋ}$
4	1 2 3	syl2an	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \lor_{ℋ} B \in C_{ℋ}$