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Theorem ontr2 6313
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 6276 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6276 . 2 (𝐶 ∈ On → Ord 𝐶)
3 ordtr2 6310 . 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 396  wcel 2106  wss 3887  Ord word 6265  Oncon0 6266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270
This theorem is referenced by:  oeordsuc  8425  oelimcl  8431  oeeui  8433  omopthlem2  8490  omxpenlem  8860  oismo  9299  cantnflem1c  9445  cantnflem1  9447  cantnflem3  9449  rankr1ai  9556  rankxplim  9637  infxpenlem  9769  alephle  9844  pwcfsdom  10339  r1limwun  10492  onelssex  33661  onunel  33689  naddssim  33837  oldbdayim  34071  ontopbas  34617  ontgval  34620
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