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Metamath Proof Explorer

Theorem onsseli

Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995)

	Ref	Expression
Hypotheses	on.1	$⊢ A \in On$
	on.2	$⊢ B \in On$
Assertion	onsseli	$⊢ A \subseteq B \leftrightarrow (A \in B \lor A = B)$

Step	Hyp	Ref	Expression
1		on.1	$⊢ A \in On$
2		on.2	$⊢ B \in On$
3		onsseleq	$⊢ (A \in On \land B \in On) \to (A \subseteq B \leftrightarrow (A \in B \lor A = B))$
4	1 2 3	mp2an	$⊢ A \subseteq B \leftrightarrow (A \in B \lor A = B)$