Theorem ontrci 6372

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 6371	. 2 ⊢ Ord 𝐴
3		ordtr 6280	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
4	2, 3	ax-mp 5	1 ⊢ Tr 𝐴

Syntax hints:   ∈ wcel 2106  Tr wtr 5191  Ord word 6265  Oncon0 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-tr 5192  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270

This theorem is referenced by:  onunisuci  6380  hfuni  34486