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Theorem ontr2 5986
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 5949 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5949 . 2 (𝐶 ∈ On → Ord 𝐶)
3 ordtr2 5983 . 2 ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
41, 2, 3syl2an 590 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wss 3767  Ord word 5938  Oncon0 5939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942  df-on 5943
This theorem is referenced by:  oeordsuc  7912  oelimcl  7918  oeeui  7920  omopthlem2  7974  omxpenlem  8301  oismo  8685  cantnflem1c  8832  cantnflem1  8834  cantnflem3  8836  rankr1ai  8909  rankxplim  8990  infxpenlem  9120  alephle  9195  pwcfsdom  9691  r1limwun  9844  ontopbas  32926  ontgval  32929
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