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Theorem trssord 4409
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 4405 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
2 wess 4380 . . . . . 6  |-  ( A 
C_  B  ->  (  _E  We  B  ->  _E  We  A ) )
32imp 418 . . . . 5  |-  ( ( A  C_  B  /\  _E  We  B )  ->  _E  We  A )
41, 3sylan2 460 . . . 4  |-  ( ( A  C_  B  /\  Ord  B )  ->  _E  We  A )
54anim2i 552 . . 3  |-  ( ( Tr  A  /\  ( A  C_  B  /\  Ord  B ) )  ->  ( Tr  A  /\  _E  We  A ) )
653impb 1147 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  _E  We  A
) )
7 df-ord 4395 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
86, 7sylibr 203 1  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    C_ wss 3152   Tr wtr 4113    _E cep 4303    We wwe 4351   Ord word 4391
This theorem is referenced by:  ordin  4422  ssorduni  4577  suceloni  4604  ordom  4665  ordtypelem2  7234  hartogs  7259  card2on  7268  tskwe  7583  ondomon  8185  dford3lem2  27120  dford3  27121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-in 3159  df-ss 3166  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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