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Theorem ontr1 4331
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 4295 . 2  |-  ( C  e.  On  ->  Ord  C )
2 ordtr1 4328 . 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 1621   Ord word 4284   Oncon0 4285
This theorem is referenced by:  smoiun  6264  dif20el  6390  oeordi  6471  omabs  6531  omsmolem  6537  cofsmo  7779  cfsmolem  7780  inar1  8277  grur1a  8321
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rex 2514  df-v 2729  df-in 3085  df-ss 3089  df-uni 3728  df-tr 4011  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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