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Theorem ordtr2 4566
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtr2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )

Proof of Theorem ordtr2
StepHypRef Expression
1 ordelord 4544 . . . . . . . 8  |-  ( ( Ord  C  /\  B  e.  C )  ->  Ord  B )
21ex 424 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  Ord  B
) )
32ancld 537 . . . . . 6  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( B  e.  C  /\  Ord  B ) ) )
43anc2li 541 . . . . 5  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
C  /\  ( B  e.  C  /\  Ord  B
) ) ) )
5 ordelpss 4550 . . . . . . . . . . 11  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  e.  C  <->  B  C.  C ) )
65ancoms 440 . . . . . . . . . 10  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  <->  B  C.  C ) )
7 sspsstr 3395 . . . . . . . . . . 11  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
87expcom 425 . . . . . . . . . 10  |-  ( B 
C.  C  ->  ( A  C_  B  ->  A  C.  C ) )
96, 8syl6bi 220 . . . . . . . . 9  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  ->  ( A 
C_  B  ->  A  C.  C ) ) )
109ex 424 . . . . . . . 8  |-  ( Ord 
C  ->  ( Ord  B  ->  ( B  e.  C  ->  ( A  C_  B  ->  A  C.  C ) ) ) )
1110com23 74 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
B  ->  ( A  C_  B  ->  A  C.  C ) ) ) )
1211imp32 423 . . . . . 6  |-  ( ( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  -> 
( A  C_  B  ->  A  C.  C ) )
1312com12 29 . . . . 5  |-  ( A 
C_  B  ->  (
( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  ->  A  C.  C
) )
144, 13syl9 68 . . . 4  |-  ( Ord 
C  ->  ( A  C_  B  ->  ( B  e.  C  ->  A  C.  C ) ) )
1514imp3a 421 . . 3  |-  ( Ord 
C  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
1615adantl 453 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
17 ordelpss 4550 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( A  e.  C  <->  A  C.  C ) )
1816, 17sylibrd 226 1  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    C_ wss 3263    C. wpss 3264   Ord word 4521
This theorem is referenced by:  ordtr3  4567  ontr2  4569  ordelinel  4620  smogt  6565  smorndom  6566  nnarcl  6795  nnawordex  6816  coftr  8086  nodenselem5  25363  hfuni  25839
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525
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