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Theorem ordtr2 4452
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 4430 . . . . . . . 8  |-  ( ( Ord  C  /\  B  e.  C )  ->  Ord  B )
21ex 423 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  Ord  B
) )
32ancld 536 . . . . . 6  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( B  e.  C  /\  Ord  B ) ) )
43anc2li 540 . . . . 5  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
C  /\  ( B  e.  C  /\  Ord  B
) ) ) )
5 ordelpss 4436 . . . . . . . . . . 11  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  e.  C  <->  B  C.  C ) )
65ancoms 439 . . . . . . . . . 10  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  <->  B  C.  C ) )
7 sspsstr 3294 . . . . . . . . . . 11  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
87expcom 424 . . . . . . . . . 10  |-  ( B 
C.  C  ->  ( A  C_  B  ->  A  C.  C ) )
96, 8syl6bi 219 . . . . . . . . 9  |-  ( ( Ord  C  /\  Ord  B )  ->  ( B  e.  C  ->  ( A 
C_  B  ->  A  C.  C ) ) )
109ex 423 . . . . . . . 8  |-  ( Ord 
C  ->  ( Ord  B  ->  ( B  e.  C  ->  ( A  C_  B  ->  A  C.  C ) ) ) )
1110com23 72 . . . . . . 7  |-  ( Ord 
C  ->  ( B  e.  C  ->  ( Ord 
B  ->  ( A  C_  B  ->  A  C.  C ) ) ) )
1211imp32 422 . . . . . 6  |-  ( ( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  -> 
( A  C_  B  ->  A  C.  C ) )
1312com12 27 . . . . 5  |-  ( A 
C_  B  ->  (
( Ord  C  /\  ( B  e.  C  /\  Ord  B ) )  ->  A  C.  C
) )
144, 13syl9 66 . . . 4  |-  ( Ord 
C  ->  ( A  C_  B  ->  ( B  e.  C  ->  A  C.  C ) ) )
1514imp3a 420 . . 3  |-  ( Ord 
C  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
1615adantl 452 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  C.  C ) )
17 ordelpss 4436 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( A  e.  C  <->  A  C.  C ) )
1816, 17sylibrd 225 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 176    /\ wa 358    e. wcel 1696    C_ wss 3165    C. wpss 3166   Ord word 4407
This theorem is referenced by:  ordtr3  4453  ontr2  4455  ordelinel  4507  smogt  6400  smorndom  6401  nnarcl  6630  nnawordex  6651  coftr  7915  nodenselem5  24410  hfuni  24886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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