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Theorem ontr1 4587
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 4551 . 2  |-  ( C  e.  On  ->  Ord  C )
2 ordtr1 4584 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   Ord word 4540   Oncon0 4541
This theorem is referenced by:  smoiun  6582  dif20el  6708  oeordi  6789  omabs  6849  omsmolem  6855  cofsmo  8105  cfsmolem  8106  inar1  8606  grur1a  8650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-uni 3976  df-tr 4263  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545
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