Theorem ontrci 4389

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

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
ontrci	⊢ Tr 𝐴

Proof of Theorem ontrci

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onordi 4388	. 2 ⊢ Ord 𝐴
3		ordtr 4340	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
4	2, 3	ax-mp 5	1 ⊢ Tr 𝐴

Syntax hints:   ∈ wcel 2128  Tr wtr 4064  Ord word 4324  Oncon0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3775  df-tr 4065  df-iord 4328  df-on 4330

This theorem is referenced by:  onunisuci  4394  exmidonfinlem  7130  bj-el2oss1o  13419  nnsf  13648