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 6373	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6377	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107  Tr wtr 5239  Ord word 6362  Oncon0 6363

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 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-v 3465  df-ss 3948  df-uni 4888  df-tr 5240  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367

This theorem is referenced by:  onunisuc  6474  ontrciOLD  6476  onuninsuci  7843  hfuni  36160