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Theorem ontr1 4092
 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 → ((A B B 𝐶) → A 𝐶))

Proof of Theorem ontr1
StepHypRef Expression
1 eloni 4078 . 2 (𝐶 On → Ord 𝐶)
2 ordtr1 4091 . 2 (Ord 𝐶 → ((A B B 𝐶) → A 𝐶))
31, 2syl 14 1 (𝐶 On → ((A B B 𝐶) → A 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Ord word 4065  Oncon0 4066 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by:  smoiun  5857  ltsopi  6304  prarloclemarch2  6402
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