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Theorem ordtr1 5731
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 5701 . 2 (Ord 𝐶 → Tr 𝐶)
2 trel 4724 . 2 (Tr 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Tr wtr 4717  Ord word 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-ss 3573  df-uni 4408  df-tr 4718  df-ord 5690
This theorem is referenced by:  ontr1  5735  dfsmo2  7396  smores2  7403  smoel  7409  smogt  7416  ordiso2  8371  r1ordg  8592  r1pwss  8598  r1val1  8600  rankr1ai  8612  rankval3b  8640  rankonidlem  8642  onssr1  8645  cofsmo  9042  fpwwe2lem9  9411  bnj1098  30589  bnj594  30717
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