Theorem trel3 4042

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 967	. . 3 |- ( ( B e. C /\ C e. D /\ D e. A ) <-> ( B e. C /\ ( C e. D /\ D e. A ) ) )
2		trel 4041	. . . 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 4041	. 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 963   e. wcel 1481   Tr wtr 4034

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035

This theorem is referenced by: (None)