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Theorem trsucss 4478
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 4458 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 trss 4122 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
3 eqimss 3230 . . . 4  |-  ( B  =  A  ->  B  C_  A )
43a1i 10 . . 3  |-  ( Tr  A  ->  ( B  =  A  ->  B  C_  A ) )
52, 4jaod 369 . 2  |-  ( Tr  A  ->  ( ( B  e.  A  \/  B  =  A )  ->  B  C_  A )
)
61, 5syl5 28 1  |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    C_ wss 3152   Tr wtr 4113   suc csuc 4394
This theorem is referenced by:  efgmnvl  15023
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-uni 3828  df-tr 4114  df-suc 4398
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