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Theorem ontr 6495
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 6396 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordtr 6400 . 2 (Ord 𝐴 → Tr 𝐴)
31, 2syl 17 1 (𝐴 ∈ On → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Tr wtr 5265  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onunisuc  6496  ontrciOLD  6498  onuninsuci  7861  hfuni  36166
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