Theorem trel3 4166

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 985	. . 3 |- ( ( B e. C /\ C e. D /\ D e. A ) <-> ( B e. C /\ ( C e. D /\ D e. A ) ) )
2		trel 4165	. . . 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 4165	. 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 981   e. wcel 2178   Tr wtr 4158

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159

This theorem is referenced by: (None)