Theorem onordi 4113

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 4061	. 2 ⊢ (A ∈ On → Ord A)
3	1, 2	ax-mp 7	1 ⊢ Ord A

Syntax hints:   ∈ wcel 1374  Ord word 4048  Oncon0 4049

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054

This theorem is referenced by:  ontrci  4114  onsucsssucexmid  4196