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Theorem ontr1 4214
Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
ontr1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Proof of Theorem ontr1
StepHypRef Expression
1 eloni 4200 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 4213 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 14 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  Ord word 4187  Oncon0 4188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3652  df-tr 3935  df-iord 4191  df-on 4193
This theorem is referenced by:  smoiun  6058  onunsnss  6617  snon0  6635  ltsopi  6869  prarloclemarch2  6968
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