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Theorem trssord 4398
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 4385 . . . . . . 7  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
21simprbi 275 . . . . . 6  |-  ( Ord 
B  ->  A. x  e.  B  Tr  x
)
3 ssralv 3234 . . . . . 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 124 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  A. x  e.  A  Tr  x
)
65anim2i 342 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  A. x  e.  A  Tr  x
) )
763impb 1201 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
8 dford3 4385 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
97, 8sylibr 134 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980   A.wral 2468    C_ wss 3144   Tr wtr 4116   Ord word 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-in 3150  df-ss 3157  df-iord 4384
This theorem is referenced by:  ordelord  4399  ordin  4403  ssorduni  4504  ordtriexmidlem  4536  ordtri2or2exmidlem  4543  onsucelsucexmidlem  4546  ordsuc  4580
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