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Metamath Proof Explorer

Theorem ordn2lp

Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of BellMachover p. 469. (Contributed by NM, 3-Apr-1994)

		Ref	Expression
	Assertion	ordn2lp	$⊢ Ord A \to \neg (A \in B \land B \in A)$

Proof

Step	Hyp	Ref	Expression
1		ordirr	$⊢ Ord A \to \neg A \in A$
2		ordtr	$⊢ Ord A \to Tr A$
3		trel	$⊢ Tr A \to ((A \in B \land B \in A) \to A \in A)$
4	2 3	syl	$⊢ Ord A \to ((A \in B \land B \in A) \to A \in A)$
5	1 4	mtod	$⊢ Ord A \to \neg (A \in B \land B \in A)$