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Theorem trelded 28331
Description: Deduction form of trel 4120. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
trelded.1  |-  ( ph  ->  Tr  A )
trelded.2  |-  ( ps 
->  B  e.  C
)
trelded.3  |-  ( ch 
->  C  e.  A
)
Assertion
Ref Expression
trelded  |-  ( (
ph  /\  ps  /\  ch )  ->  B  e.  A
)

Proof of Theorem trelded
StepHypRef Expression
1 trelded.1 . 2  |-  ( ph  ->  Tr  A )
2 trelded.2 . 2  |-  ( ps 
->  B  e.  C
)
3 trelded.3 . 2  |-  ( ch 
->  C  e.  A
)
4 trel 4120 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
543impib 1149 . 2  |-  ( ( Tr  A  /\  B  e.  C  /\  C  e.  A )  ->  B  e.  A )
61, 2, 3, 5syl3an 1224 1  |-  ( (
ph  /\  ps  /\  ch )  ->  B  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684   Tr wtr 4113
This theorem is referenced by:  suctrALT3  28700
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
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