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Metamath Proof Explorer

Theorem shjval

Description: Value of join in SH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

		Ref	Expression
	Assertion	shjval	$⊢ (A \in S_{ℋ} \land B \in S_{ℋ}) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$

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		sshjval	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$
4	1 2 3	syl2an	$⊢ (A \in S_{ℋ} \land B \in S_{ℋ}) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$