Theorem ontr 6500

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 6398	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6402	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Tr wtr 5218  Ord word 6387  Oncon0 6388

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4873  df-tr 5219  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392

This theorem is used by:  onunisuc  6501  onuninsuci  7885  hfuniOLD  9962  nmuladdss  37177  par2catid  51098