Theorem ontr 6388

Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Assertion

Ref	Expression
ontr	⊢ (𝐴 ∈ On → Tr 𝐴)

Proof of Theorem ontr

Step	Hyp	Ref	Expression
1		eloni 6291	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordtr 6295	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ On → Tr 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2104  Tr wtr 5198  Ord word 6280  Oncon0 6281

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3439  df-in 3899  df-ss 3909  df-uni 4845  df-tr 5199  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-ord 6284  df-on 6285

This theorem is referenced by:  onunisuc  6389  ontrci  6391