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Theorem trssord 4143
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 4130 . . . . . . 7  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
21simprbi 269 . . . . . 6  |-  ( Ord 
B  ->  A. x  e.  B  Tr  x
)
3 ssralv 3059 . . . . . 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 1135 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
8 dford3 4130 . 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 920   A.wral 2349    C_ wss 2974   Tr wtr 3883   Ord word 4125
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987  df-iord 4129
This theorem is referenced by:  ordelord  4144  ordin  4148  ssorduni  4239  ordtriexmidlem  4271  ordtri2or2exmidlem  4277  onsucelsucexmidlem  4280  ordsuc  4314
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