Theorem ontrci 6128

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 6127	. 2 ⊢ Ord 𝐴
3		ordtr 6037	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
4	2, 3	ax-mp 5	1 ⊢ Tr 𝐴

Syntax hints:   ∈ wcel 2048  Tr wtr 5024  Ord word 6022  Oncon0 6023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-in 3832  df-ss 3839  df-uni 4707  df-tr 5025  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-ord 6026  df-on 6027

This theorem is referenced by:  onunisuci  6136  hfuni  33106