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Theorem trsucss 5508
Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
trsucss  |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A ) )

Proof of Theorem trsucss
StepHypRef Expression
1 elsuci 5489 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 trss 4506 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
3 eqimss 3484 . . . 4  |-  ( B  =  A  ->  B  C_  A )
43a1i 11 . . 3  |-  ( Tr  A  ->  ( B  =  A  ->  B  C_  A ) )
52, 4jaod 382 . 2  |-  ( Tr  A  ->  ( ( B  e.  A  \/  B  =  A )  ->  B  C_  A )
)
61, 5syl5 33 1  |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    = wceq 1444    e. wcel 1887    C_ wss 3404   Tr wtr 4497   suc csuc 5425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-v 3047  df-un 3409  df-in 3411  df-ss 3418  df-sn 3969  df-uni 4199  df-tr 4498  df-suc 5429
This theorem is referenced by:  efgmnvl  17364
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