Theorem onordi 4129

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

Hypothesis

Ref	Expression
on.1	⊢ A ∈ On

Assertion

Ref	Expression
onordi	⊢ Ord A

Proof of Theorem onordi

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ A ∈ On
2		eloni 4078	. 2 ⊢ (A ∈ On → Ord A)
3	1, 2	ax-mp 7	1 ⊢ Ord A

Syntax hints:   ∈ wcel 1390  Ord word 4065  Oncon0 4066

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071

This theorem is referenced by:  ontrci  4130  onsucsssucexmid  4212