Theorem epne3

Description: A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994) (Revised by Mario Carneiro, 22-Jun-2015)

	Ref	Expression
Assertion	epne3	|- ( ( _E Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B e. C /\ C e. D /\ D e. B ) )

Proof

Step	Hyp	Ref	Expression
1		fr3nr	|- ( ( _E Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B _E C /\ C _E D /\ D _E B ) )
2		epelg	|- ( C e. A -> ( B _E C <-> B e. C ) )
3	2	3ad2ant2	|- ( ( B e. A /\ C e. A /\ D e. A ) -> ( B _E C <-> B e. C ) )
4		epelg	|- ( D e. A -> ( C _E D <-> C e. D ) )
5	4	3ad2ant3	|- ( ( B e. A /\ C e. A /\ D e. A ) -> ( C _E D <-> C e. D ) )
6		epelg	|- ( B e. A -> ( D _E B <-> D e. B ) )
7	6	3ad2ant1	|- ( ( B e. A /\ C e. A /\ D e. A ) -> ( D _E B <-> D e. B ) )
8	3 5 7	3anbi123d	|- ( ( B e. A /\ C e. A /\ D e. A ) -> ( ( B _E C /\ C _E D /\ D _E B ) <-> ( B e. C /\ C e. D /\ D e. B ) ) )
9	8	adantl	|- ( ( _E Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B _E C /\ C _E D /\ D _E B ) <-> ( B e. C /\ C e. D /\ D e. B ) ) )
10	1 9	mtbid	|- ( ( _E Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B e. C /\ C e. D /\ D e. B ) )