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Theorem ordtr1 5805
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 5775 . 2 (Ord 𝐶 → Tr 𝐶)
2 trel 4792 . 2 (Tr 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Tr wtr 4785  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786  df-ord 5764
This theorem is referenced by:  ontr1  5809  dfsmo2  7489  smores2  7496  smoel  7502  smogt  7509  ordiso2  8461  r1ordg  8679  r1pwss  8685  r1val1  8687  rankr1ai  8699  rankval3b  8727  rankonidlem  8729  onssr1  8732  cofsmo  9129  fpwwe2lem9  9498  bnj1098  30980  bnj594  31108  nosepssdm  31961
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