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Metamath Proof Explorer

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 ) )