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Theorem trsucss 4601
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 4582 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 trss 4246 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
3 eqimss 3337 . . . 4  |-  ( B  =  A  ->  B  C_  A )
43a1i 11 . . 3  |-  ( Tr  A  ->  ( B  =  A  ->  B  C_  A ) )
52, 4jaod 370 . 2  |-  ( Tr  A  ->  ( ( B  e.  A  \/  B  =  A )  ->  B  C_  A )
)
61, 5syl5 30 1  |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1717    C_ wss 3257   Tr wtr 4237   suc csuc 4518
This theorem is referenced by:  efgmnvl  15267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-v 2895  df-un 3262  df-in 3264  df-ss 3271  df-sn 3757  df-uni 3952  df-tr 4238  df-suc 4522
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