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Theorem trel3 3903
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 924 . . 3  |-  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  <->  ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) ) )
2 trel 3902 . . . 4  |-  ( Tr  A  ->  ( ( C  e.  D  /\  D  e.  A )  ->  C  e.  A ) )
32anim2d 330 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) )  ->  ( B  e.  C  /\  C  e.  A )
) )
41, 3syl5bi 150 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  ( B  e.  C  /\  C  e.  A
) ) )
5 trel 3902 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
64, 5syld 44 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 102    /\ w3a 920    e. wcel 1434   Tr wtr 3895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896
This theorem is referenced by: (None)
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