Theorem ontr 6417

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 6316	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6320	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2111  Tr wtr 5196  Ord word 6305  Oncon0 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-ss 3914  df-uni 4857  df-tr 5197  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310

This theorem is referenced by:  onunisuc  6418  onuninsuci  7770  hfuni  36226