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Theorem ontr1 6402
Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
ontr1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Proof of Theorem ontr1
StepHypRef Expression
1 eloni 6366 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 6399 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Ord word 6355  Oncon0 6356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3953  df-ss 3963  df-uni 4905  df-tr 5262  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6359  df-on 6360
This theorem is referenced by:  epweon  7749  smoiun  8348  dif20el  8492  oeordi  8575  omabs  8638  omsmolem  8644  naddel12  8687  cofsmo  10251  cfsmolem  10252  inar1  10757  grur1a  10801  nosupno  27173  nosupbnd2lem1  27185  noinfno  27188  noinfbnd2lem1  27200  lrrecpo  27392  addsproplem2  27421  onexoegt  41864  oneltr  41876  oaun3lem1  41995  nadd2rabtr  42005  naddsuc2  42014  naddwordnexlem0  42018  oawordex3  42022  naddwordnexlem4  42023
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