Theorem efrn2lp

Description: A well-founded class contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994)

	Ref	Expression
Assertion	efrn2lp	|- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B e. C /\ C e. B ) )

Proof

Step	Hyp	Ref	Expression
1		fr2nr	|- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B _E C /\ C _E B ) )
2		epelg	|- ( C e. A -> ( B _E C <-> B e. C ) )
3		epelg	|- ( B e. A -> ( C _E B <-> C e. B ) )
4	2 3	bi2anan9r	|- ( ( B e. A /\ C e. A ) -> ( ( B _E C /\ C _E B ) <-> ( B e. C /\ C e. B ) ) )
5	4	adantl	|- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> ( ( B _E C /\ C _E B ) <-> ( B e. C /\ C e. B ) ) )
6	1 5	mtbid	|- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B e. C /\ C e. B ) )