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Theorem ordtr2 4912
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 4890 . . . . . . . 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 4896 . . . . . . . . . . 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 3594 . . . . . . . . . . 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 4896 . 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 1804    C_ wss 3461    C. wpss 3462   Ord word 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871
This theorem is referenced by:  ordtr3  4913  ontr2  4915  ordelinel  4966  smogt  7040  smorndom  7041  nnarcl  7267  nnawordex  7288  coftr  8656  nodenselem5  29420  hfuni  29816
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