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Theorem ontr1 4438
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 4402 . 2  |-  ( C  e.  On  ->  Ord  C )
2 ordtr1 4435 . 2  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 15 1  |-  ( C  e.  On  ->  (
( A  e.  B  /\  B  e.  C
)  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   Ord word 4391   Oncon0 4392
This theorem is referenced by:  smoiun  6378  dif20el  6504  oeordi  6585  omabs  6645  omsmolem  6651  cofsmo  7895  cfsmolem  7896  inar1  8397  grur1a  8441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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