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Theorem ontr1 6259
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 6223 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 6256 . 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 2110  Ord word 6212  Oncon0 6213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-tr 5162  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217
This theorem is referenced by:  smoiun  8098  dif20el  8232  oeordi  8315  omabs  8376  omsmolem  8382  cofsmo  9883  cfsmolem  9884  inar1  10389  grur1a  10433  nosupno  33643  nosupbnd2lem1  33655  noinfno  33658  noinfbnd2lem1  33670  lrrecpo  33835
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