MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trsucss Structured version   Unicode version

Theorem trsucss 4660
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 4640 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 trss 4304 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
3 eqimss 3393 . . . 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 1652    e. wcel 1725    C_ wss 3313   Tr wtr 4295   suc csuc 4576
This theorem is referenced by:  efgmnvl  15339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-v 2951  df-un 3318  df-in 3320  df-ss 3327  df-sn 3813  df-uni 4009  df-tr 4296  df-suc 4580
  Copyright terms: Public domain W3C validator