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Theorem ontr2 5670
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 5631 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5631 . 2 (𝐶 ∈ On → Ord 𝐶)
3 ordtr2 5666 . 2 ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
41, 2, 3syl2an 492 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1975  wss 3534  Ord word 5620  Oncon0 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-tr 4670  df-eprel 4934  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-ord 5624  df-on 5625
This theorem is referenced by:  oeordsuc  7533  oelimcl  7539  oeeui  7541  omopthlem2  7595  omxpenlem  7918  oismo  8300  cantnflem1c  8439  cantnflem1  8441  cantnflem3  8443  rankr1ai  8516  rankxplim  8597  infxpenlem  8691  alephle  8766  pwcfsdom  9256  r1limwun  9409  nobndlem6  30897  ontopbas  31398  ontgval  31401
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