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Theorem ontr2 6237
 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 6200 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6200 . 2 (𝐶 ∈ On → Ord 𝐶)
3 ordtr2 6234 . 2 ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
41, 2, 3syl2an 595 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2107   ⊆ wss 3940  Ord word 6189  Oncon0 6190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194 This theorem is referenced by:  oeordsuc  8215  oelimcl  8221  oeeui  8223  omopthlem2  8278  omxpenlem  8612  oismo  8998  cantnflem1c  9144  cantnflem1  9146  cantnflem3  9148  rankr1ai  9221  rankxplim  9302  infxpenlem  9433  alephle  9508  pwcfsdom  9999  r1limwun  10152  ontopbas  33679  ontgval  33682
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