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Theorem trel3 4251
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
StepHypRef Expression
1 3anass 940 . . 3  |-  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  <->  ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) ) )
2 trel 4250 . . . 4  |-  ( Tr  A  ->  ( ( C  e.  D  /\  D  e.  A )  ->  C  e.  A ) )
32anim2d 549 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) )  ->  ( B  e.  C  /\  C  e.  A )
) )
41, 3syl5bi 209 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  ( B  e.  C  /\  C  e.  A
) ) )
5 trel 4250 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
64, 5syld 42 1  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1717   Tr wtr 4243
This theorem is referenced by:  ordelord  4544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-ss 3277  df-uni 3958  df-tr 4244
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