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Theorem ontr1 4439
Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
ontr1  |-  ( C  e.  On  ->  (
( A  e.  B  /\  B  e.  C
)  ->  A  e.  C ) )

Proof of Theorem ontr1
StepHypRef Expression
1 eloni 4403 . 2  |-  ( C  e.  On  ->  Ord  C )
2 ordtr1 4436 . 2  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 17 1  |-  ( C  e.  On  ->  (
( A  e.  B  /\  B  e.  C
)  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1687   Ord word 4392   Oncon0 4393
This theorem is referenced by:  smoiun  6375  dif20el  6501  oeordi  6582  omabs  6642  omsmolem  6648  cofsmo  7892  cfsmolem  7893  inar1  8394  grur1a  8438
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ral 2551  df-rex 2552  df-v 2793  df-in 3162  df-ss 3169  df-uni 3831  df-tr 4117  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397
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