Theorem ontrci 6264

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 6263	. 2 ⊢ Ord 𝐴
3		ordtr 6173	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
4	2, 3	ax-mp 5	1 ⊢ Tr 𝐴

Syntax hints:   ∈ wcel 2111  Tr wtr 5136  Ord word 6158  Oncon0 6159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163

This theorem is referenced by:  onunisuci  6272  hfuni  33758