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Metamath Proof Explorer

Theorem trel

Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994) (Proof shortened by Andrew Salmon, 9-Jul-2011)

Ref Expression
Assertion trel 
|- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )

Proof

Step Hyp Ref Expression
1 dftr2 
 |-  ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2 eleq12 
 |-  ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3 eleq1 
 |-  ( x = C -> ( x e. A <-> C e. A ) )
4 3 adantl 
 |-  ( ( y = B /\ x = C ) -> ( x e. A <-> C e. A ) )
5 2 4 anbi12d 
 |-  ( ( y = B /\ x = C ) -> ( ( y e. x /\ x e. A ) <-> ( B e. C /\ C e. A ) ) )
6 eleq1 
 |-  ( y = B -> ( y e. A <-> B e. A ) )
7 6 adantr 
 |-  ( ( y = B /\ x = C ) -> ( y e. A <-> B e. A ) )
8 5 7 imbi12d 
 |-  ( ( y = B /\ x = C ) -> ( ( ( y e. x /\ x e. A ) -> y e. A ) <-> ( ( B e. C /\ C e. A ) -> B e. A ) ) )
9 8 spc2gv 
 |-  ( ( B e. C /\ C e. A ) -> ( A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) -> ( ( B e. C /\ C e. A ) -> B e. A ) ) )
10 9 pm2.43b 
 |-  ( A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) -> ( ( B e. C /\ C e. A ) -> B e. A ) )
11 1 10 sylbi 
 |-  ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )