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Theorem trelded 36362
Description: Deduction form of trel 4496. 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 4496 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
543impib 1195 . 2  |-  ( ( Tr  A  /\  B  e.  C  /\  C  e.  A )  ->  B  e.  A )
61, 2, 3, 5syl3an 1272 1  |-  ( (
ph  /\  ps  /\  ch )  ->  B  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    e. wcel 1842   Tr wtr 4489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-in 3421  df-ss 3428  df-uni 4192  df-tr 4490
This theorem is referenced by:  suctrALT3  36755
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