Theorem trel3 4195

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 1008	. . 3 |- ( ( B e. C /\ C e. D /\ D e. A ) <-> ( B e. C /\ ( C e. D /\ D e. A ) ) )
2		trel 4194	. . . 4 |- ( Tr A -> ( ( C e. D /\ D e. A ) -> C e. A ) )
3	2	anim2d 337	. . 3 |- ( Tr A -> ( ( B e. C /\ ( C e. D /\ D e. A ) ) -> ( B e. C /\ C e. A ) ) )
4	1, 3	biimtrid 152	. 2 |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> ( B e. C /\ C e. A ) ) )
5		trel 4194	. 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 104   /\ w3a 1004   e. wcel 2202   Tr wtr 4187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188

This theorem is referenced by: (None)