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Theorem trelded 28652
Description: Deduction form of trel 4309. 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 4309 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
543impib 1151 . 2  |-  ( ( Tr  A  /\  B  e.  C  /\  C  e.  A )  ->  B  e.  A )
61, 2, 3, 5syl3an 1226 1  |-  ( (
ph  /\  ps  /\  ch )  ->  B  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1725   Tr wtr 4302
This theorem is referenced by:  suctrALT3  29036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-uni 4016  df-tr 4303
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