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Metamath Proof Explorer

Theorem orddisj

Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998)

		Ref	Expression
	Assertion	orddisj	$⊢ Ord A \to A \cap \{A\} = \emptyset$

Step	Hyp	Ref	Expression
1		ordirr	$⊢ Ord A \to \neg A \in A$
2		disjsn	$⊢ A \cap \{A\} = \emptyset \leftrightarrow \neg A \in A$
3	1 2	sylibr	$⊢ Ord A \to A \cap \{A\} = \emptyset$