ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  trsucss Unicode version

Theorem trsucss 4250
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 4230 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 trss 3945 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
3 eqimss 3078 . . . 4  |-  ( B  =  A  ->  B  C_  A )
43a1i 9 . . 3  |-  ( Tr  A  ->  ( B  =  A  ->  B  C_  A ) )
52, 4jaod 672 . 2  |-  ( Tr  A  ->  ( ( B  e.  A  \/  B  =  A )  ->  B  C_  A )
)
61, 5syl5 32 1  |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664    = wceq 1289    e. wcel 1438    C_ wss 2999   Tr wtr 3936   suc csuc 4192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-uni 3654  df-tr 3937  df-suc 4198
This theorem is referenced by:  onsucsssucr  4326  ordpwsucss  4383  nnsf  11895  nninfalllemn  11898
  Copyright terms: Public domain W3C validator