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Theorem ordtr2 4436
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 4414 . . . . . . . 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 4420 . . . . . . . . . . 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 3281 . . . . . . . . . . 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 4420 . 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 1684    C_ wss 3152    C. wpss 3153   Ord word 4391
This theorem is referenced by:  ordtr3  4437  ontr2  4439  ordelinel  4491  smogt  6384  smorndom  6385  nnarcl  6614  nnawordex  6635  coftr  7899  nodenselem5  24339  hfuni  24814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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