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Theorem trel3 4083
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 971 . . 3  |-  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  <->  ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) ) )
2 trel 4082 . . . 4  |-  ( Tr  A  ->  ( ( C  e.  D  /\  D  e.  A )  ->  C  e.  A ) )
32anim2d 335 . . 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 4082 . 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 967    e. wcel 2135   Tr wtr 4075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-in 3118  df-ss 3125  df-uni 3785  df-tr 4076
This theorem is referenced by: (None)
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