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

Theorem trssord 4205
Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
Assertion
Ref Expression
trssord  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )

Proof of Theorem trssord
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dford3 4192 . . . . . . 7  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
21simprbi 269 . . . . . 6  |-  ( Ord 
B  ->  A. x  e.  B  Tr  x
)
3 ssralv 3085 . . . . . 6  |-  ( A 
C_  B  ->  ( A. x  e.  B  Tr  x  ->  A. x  e.  A  Tr  x
) )
42, 3syl5 32 . . . . 5  |-  ( A 
C_  B  ->  ( Ord  B  ->  A. x  e.  A  Tr  x
) )
54imp 122 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  A. x  e.  A  Tr  x
)
65anim2i 334 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  A. x  e.  A  Tr  x
) )
763impb 1139 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
8 dford3 4192 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
97, 8sylibr 132 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924   A.wral 2359    C_ wss 2999   Tr wtr 3934   Ord word 4187
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  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-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-in 3005  df-ss 3012  df-iord 4191
This theorem is referenced by:  ordelord  4206  ordin  4210  ssorduni  4302  ordtriexmidlem  4334  ordtri2or2exmidlem  4340  onsucelsucexmidlem  4343  ordsuc  4377
  Copyright terms: Public domain W3C validator