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Theorem trssord 4847
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 4843 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
2 wess 4818 . . . . . 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 1184 . 2  |-  ( ( Tr  A  /\  A  C_  B  /\  Ord  B
)  ->  ( Tr  A  /\  _E  We  A
) )
7 df-ord 4833 . 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 965    C_ wss 3439   Tr wtr 4496    _E cep 4741    We wwe 4789   Ord word 4829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-in 3446  df-ss 3453  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833
This theorem is referenced by:  ordin  4860  ssorduni  6510  suceloni  6537  ordom  6598  ordtypelem2  7848  hartogs  7873  card2on  7884  tskwe  8235  ondomon  8842  dford3lem2  29547  dford3  29548
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