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Metamath Proof Explorer

Theorem shjcl

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

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

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