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Metamath Proof Explorer

Theorem onssneli

Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994)

		Ref	Expression
	Hypothesis	on.1	$⊢ A \in On$
	Assertion	onssneli	$⊢ A \subseteq B \to \neg B \in A$

Step	Hyp	Ref	Expression
1		on.1	$⊢ A \in On$
2		ssel	$⊢ A \subseteq B \to (B \in A \to B \in B)$
3	1	oneli	$⊢ B \in A \to B \in On$
4		eloni	$⊢ B \in On \to Ord B$
5		ordirr	$⊢ Ord B \to \neg B \in B$
6	3 4 5	3syl	$⊢ B \in A \to \neg B \in B$
7	2 6	nsyli	$⊢ A \subseteq B \to (B \in A \to \neg B \in A)$
8	7	pm2.01d	$⊢ A \subseteq B \to \neg B \in A$