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Theorem ordtr1 4153
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 4143 . 2 (Ord 𝐶 → Tr 𝐶)
2 trel 3889 . 2 (Tr 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 14 1 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Tr wtr 3882  Ord word 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131
This theorem is referenced by:  ontr1  4154  ordwe  4328  dfsmo2  5933  smores2  5940  smoel  5946  ordiso2  6415
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