Theorem ontr 6476

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 6374	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6378	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Tr wtr 5220  Ord word 6363  Oncon0 6364

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-uni 4875  df-tr 5221  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368

This theorem is used by:  onunisuc  6477  onuninsuci  7842  hfuni  36715  nmuladdss  36744