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Theorem trelded 36926
Description: Deduction form of trel 4503. 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 4503 . . 3  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
543impib 1205 . 2  |-  ( ( Tr  A  /\  B  e.  C  /\  C  e.  A )  ->  B  e.  A )
61, 2, 3, 5syl3an 1309 1  |-  ( (
ph  /\  ps  /\  ch )  ->  B  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 984    e. wcel 1886   Tr wtr 4496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-in 3410  df-ss 3417  df-uni 4198  df-tr 4497
This theorem is referenced by:  suctrALT3  37315
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