Theorem ontr 6473

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 6371	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6375	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2149  Tr wtr 5222  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-ss 3930  df-uni 4877  df-tr 5223  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  onunisuc  6474  onuninsuci  7836  hfuni  36575