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Theorem ontr1 6206
 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 6170 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 6203 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Ord word 6159  Oncon0 6160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802  df-tr 5138  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-ord 6163  df-on 6164 This theorem is referenced by:  smoiun  7984  dif20el  8116  oeordi  8199  omabs  8260  omsmolem  8266  cofsmo  9683  cfsmolem  9684  inar1  10189  grur1a  10233  nosupno  33331  nosupbnd2lem1  33343
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