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Theorem trssord 4901
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 4897 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
2 wess 4872 . . . . . 6  |-  ( A 
C_  B  ->  (  _E  We  B  ->  _E  We  A ) )
32imp 429 . . . . 5  |-  ( ( A  C_  B  /\  _E  We  B )  ->  _E  We  A )
41, 3sylan2 474 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  _E  We  A )
54anim2i 569 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  _E  We  A ) )
653impb 1192 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  _E  We  A
) )
7 df-ord 4887 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
86, 7sylibr 212 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    C_ wss 3481   Tr wtr 4546    _E cep 4795    We wwe 4843   Ord word 4883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2822  df-in 3488  df-ss 3495  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887
This theorem is referenced by:  ordin  4914  ssorduni  6616  suceloni  6643  ordom  6704  ordtypelem2  7956  hartogs  7981  card2on  7992  tskwe  8343  ondomon  8950  dford3lem2  30897  dford3  30898
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