Theorem trel3 4095

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 977	. . 3 |- ( ( B e. C /\ C e. D /\ D e. A ) <-> ( B e. C /\ ( C e. D /\ D e. A ) ) )
2		trel 4094	. . . 4 |- ( Tr A -> ( ( C e. D /\ D e. A ) -> C e. A ) )
3	2	anim2d 335	. . 3 |- ( Tr A -> ( ( B e. C /\ ( C e. D /\ D e. A ) ) -> ( B e. C /\ C e. A ) ) )
4	1, 3	syl5bi 151	. 2 |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> ( B e. C /\ C e. A ) ) )
5		trel 4094	. 2 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )
6	4, 5	syld 45	1 |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   e. wcel 2141   Tr wtr 4087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088

This theorem is referenced by: (None)