Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ontr1 GIF version

Theorem ontr1 4126
 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 4112 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 4125 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 14 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  Ord word 4099  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  smoiun  5916  onunsnss  6355  snon0  6356  ltsopi  6418  prarloclemarch2  6517
 Copyright terms: Public domain W3C validator