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Theorem trssord 4590
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 4586 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
2 wess 4561 . . . . . 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 4576 . 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 3312   Tr wtr 4294    _E cep 4484    We wwe 4532   Ord word 4572
This theorem is referenced by:  ordin  4603  ssorduni  4758  suceloni  4785  ordom  4846  ordtypelem2  7480  hartogs  7505  card2on  7514  tskwe  7829  ondomon  8430  dford3lem2  27089  dford3  27090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-in 3319  df-ss 3326  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
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