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Theorem ontr 6492
Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.)
Assertion
Ref Expression
ontr (𝐴 ∈ On → Tr 𝐴)

Proof of Theorem ontr
StepHypRef Expression
1 eloni 6393 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordtr 6397 . 2 (Ord 𝐴 → Tr 𝐴)
31, 2syl 17 1 (𝐴 ∈ On → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Tr wtr 5258  Ord word 6382  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-ss 3967  df-uni 4907  df-tr 5259  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  onunisuc  6493  ontrciOLD  6495  onuninsuci  7862  hfuni  36186
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