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Theorem trel3 3966
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 931 . . 3  |-  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  <->  ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) ) )
2 trel 3965 . . . 4  |-  ( Tr  A  ->  ( ( C  e.  D  /\  D  e.  A )  ->  C  e.  A ) )
32anim2d 331 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) )  ->  ( B  e.  C  /\  C  e.  A )
) )
41, 3syl5bi 151 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  ( B  e.  C  /\  C  e.  A
) ) )
5 trel 3965 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
64, 5syld 45 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 103    /\ w3a 927    e. wcel 1445   Tr wtr 3958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959
This theorem is referenced by: (None)
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