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Theorem trssord 4541
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
StepHypRef Expression
1 ordwe 4537 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
2 wess 4512 . . . . . 6  |-  ( A 
C_  B  ->  (  _E  We  B  ->  _E  We  A ) )
32imp 419 . . . . 5  |-  ( ( A  C_  B  /\  _E  We  B )  ->  _E  We  A )
41, 3sylan2 461 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  _E  We  A )
54anim2i 553 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  _E  We  A ) )
653impb 1149 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  _E  We  A
) )
7 df-ord 4527 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
86, 7sylibr 204 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    C_ wss 3265   Tr wtr 4245    _E cep 4435    We wwe 4483   Ord word 4523
This theorem is referenced by:  ordin  4554  ssorduni  4708  suceloni  4735  ordom  4796  ordtypelem2  7423  hartogs  7448  card2on  7457  tskwe  7772  ondomon  8373  dford3lem2  26791  dford3  26792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-in 3272  df-ss 3279  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527
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