MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trel3 Unicode version

Theorem trel3 4121
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 938 . . 3  |-  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  <->  ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) ) )
2 trel 4120 . . . 4  |-  ( Tr  A  ->  ( ( C  e.  D  /\  D  e.  A )  ->  C  e.  A ) )
32anim2d 548 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  ( C  e.  D  /\  D  e.  A
) )  ->  ( B  e.  C  /\  C  e.  A )
) )
41, 3syl5bi 208 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  ( B  e.  C  /\  C  e.  A
) ) )
5 trel 4120 . 2  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
64, 5syld 40 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 358    /\ w3a 934    e. wcel 1684   Tr wtr 4113
This theorem is referenced by:  ordelord  4414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114
  Copyright terms: Public domain W3C validator