Theorem ontr 6469

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

Step	Hyp	Ref	Expression
1		eloni 6367	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6371	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Tr wtr 5212  Ord word 6356  Oncon0 6357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361

This theorem is used by:  onunisuc  6470  onuninsuci  7837  hfuni  36765  nmuladdss  36794