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Theorem ontr2 6431
Description: Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
ontr2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Proof of Theorem ontr2
StepHypRef Expression
1 eloni 6394 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6394 . 2 (𝐶 ∈ On → Ord 𝐶)
3 ordtr2 6428 . 2 ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
41, 2, 3syl2an 596 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wss 3951  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  onelssex  6432  onunel  6489  oeordsuc  8632  oelimcl  8638  oeeui  8640  omopthlem2  8698  coflton  8709  cofon1  8710  cofon2  8711  naddssim  8723  omxpenlem  9113  oismo  9580  cantnflem1c  9727  cantnflem1  9729  cantnflem3  9731  rankr1ai  9838  rankxplim  9919  infxpenlem  10053  alephle  10128  pwcfsdom  10623  r1limwun  10776  oldbdayim  27927  addsbdaylem  28049  negsbdaylem  28088  ontopbas  36429  ontgval  36432  onexlimgt  43255  nnoeomeqom  43325  omabs2  43345  oaun3lem2  43388  nadd2rabex  43399  nadd1suc  43405
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