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

Theorem trssord 4297
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 4284 . . . . . . 7  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
21simprbi 273 . . . . . 6  |-  ( Ord 
B  ->  A. x  e.  B  Tr  x
)
3 ssralv 3156 . . . . . 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 123 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  A. x  e.  A  Tr  x
)
65anim2i 339 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  A. x  e.  A  Tr  x
) )
763impb 1177 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
8 dford3 4284 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
97, 8sylibr 133 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962   A.wral 2414    C_ wss 3066   Tr wtr 4021   Ord word 4279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-in 3072  df-ss 3079  df-iord 4283
This theorem is referenced by:  ordelord  4298  ordin  4302  ssorduni  4398  ordtriexmidlem  4430  ordtri2or2exmidlem  4436  onsucelsucexmidlem  4439  ordsuc  4473
  Copyright terms: Public domain W3C validator