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Theorem ontr1 4874
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 4838 . 2  |-  ( C  e.  On  ->  Ord  C )
2 ordtr1 4871 . 2  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 16 1  |-  ( C  e.  On  ->  (
( A  e.  B  /\  B  e.  C
)  ->  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   Ord word 4827   Oncon0 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3444  df-ss 3451  df-uni 4201  df-tr 4495  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832
This theorem is referenced by:  smoiun  6933  dif20el  7056  oeordi  7137  omabs  7197  omsmolem  7203  cofsmo  8550  cfsmolem  8551  inar1  9054  grur1a  9098
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