Theorem ontr 6468

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 17	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  Tr wtr 5234  Ord word 6356  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-v 3466  df-ss 3948  df-uni 4889  df-tr 5235  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by:  onunisuc  6469  ontrciOLD  6471  onuninsuci  7840  hfuni  36207