Theorem trel3 3883

Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)

Assertion

Ref	Expression
trel3	|- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) )

Proof of Theorem trel3

Step	Hyp	Ref	Expression
1		3anass 923	. . 3 |- ( ( B e. C /\ C e. D /\ D e. A ) <-> ( B e. C /\ ( C e. D /\ D e. A ) ) )
2		trel 3882	. . . 4 |- ( Tr A -> ( ( C e. D /\ D e. A ) -> C e. A ) )
3	2	anim2d 330	. . 3 |- ( Tr A -> ( ( B e. C /\ ( C e. D /\ D e. A ) ) -> ( B e. C /\ C e. A ) ) )
4	1, 3	syl5bi 150	. 2 |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> ( B e. C /\ C e. A ) ) )
5		trel 3882	. 2 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )
6	4, 5	syld 44	1 |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   e. wcel 1433   Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876

This theorem is referenced by: (None)