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Theorem ordtr2 4922
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 4900 . . . . . . . 8  |-  ( ( Ord  C  /\  B  e.  C )  ->  Ord  B )
21ex 434 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  Ord  B
) )
32ancld 553 . . . . . 6  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( B  e.  C  /\  Ord  B ) ) )
43anc2li 557 . . . . 5  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
C  /\  ( B  e.  C  /\  Ord  B
) ) ) )
5 ordelpss 4906 . . . . . . . . . . 11  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  e.  C  <->  B  C.  C ) )
65ancoms 453 . . . . . . . . . 10  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  <->  B  C.  C ) )
7 sspsstr 3609 . . . . . . . . . . 11  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
87expcom 435 . . . . . . . . . 10  |-  ( B 
C.  C  ->  ( A  C_  B  ->  A  C.  C ) )
96, 8syl6bi 228 . . . . . . . . 9  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  ->  ( A 
C_  B  ->  A  C.  C ) ) )
109ex 434 . . . . . . . 8  |-  ( Ord 
C  ->  ( Ord  B  ->  ( B  e.  C  ->  ( A  C_  B  ->  A  C.  C
) ) ) )
1110com23 78 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
B  ->  ( A  C_  B  ->  A  C.  C
) ) ) )
1211imp32 433 . . . . . 6  |-  ( ( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  -> 
( A  C_  B  ->  A  C.  C )
)
1312com12 31 . . . . 5  |-  ( A 
C_  B  ->  (
( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  ->  A  C.  C
) )
144, 13syl9 71 . . . 4  |-  ( Ord 
C  ->  ( A  C_  B  ->  ( B  e.  C  ->  A  C.  C ) ) )
1514impd 431 . . 3  |-  ( Ord 
C  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
1615adantl 466 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
17 ordelpss 4906 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( A  e.  C  <->  A  C.  C ) )
1816, 17sylibrd 234 1  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3476    C. wpss 3477   Ord word 4877
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881
This theorem is referenced by:  ordtr3  4923  ontr2  4925  ordelinel  4976  smogt  7038  smorndom  7039  nnarcl  7265  nnawordex  7286  coftr  8652  nodenselem5  29038  hfuni  29434
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