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Theorem ordtr1 4280
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
ordtr1 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Proof of Theorem ordtr1
StepHypRef Expression
1 ordtr 4270 . 2 (Ord 𝐶 → Tr 𝐶)
2 trel 4003 . 2 (Tr 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 14 1 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  Tr wtr 3996  Ord word 4254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258
This theorem is referenced by:  ontr1  4281  ordwe  4460  dfsmo2  6152  smores2  6159  smoel  6165  tfr1onlemsucaccv  6206  tfr1onlembxssdm  6208  tfr1onlembfn  6209  tfr1onlemaccex  6213  tfr1onlemres  6214  tfrcllemsucaccv  6219  tfrcllembxssdm  6221  tfrcllembfn  6222  tfrcllemaccex  6226  tfrcllemres  6227  tfrcl  6229  ordiso2  6888
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