Theorem onordi 6436

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 6333	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ord 𝐴

Syntax hints:   ∈ wcel 2114  Ord word 6322  Oncon0 6323

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-ss 3906  df-uni 4851  df-tr 5193  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  onirri  6437  onsucssi  7792  ord1eln01  8431  ord2eln012  8432  oawordeulem  8489  omopthi  8597  en2  9190  en3  9191  ssttrcl  9636  ttrcltr  9637  dmttrcl  9642  ttrclselem2  9647  bndrank  9765  rankprb  9775  rankuniss  9790  rankelun  9796  rankelpr  9797  rankelop  9798  rankmapu  9802  rankxplim3  9805  rankxpsuc  9806  cardlim  9896  carduni  9905  dfac8b  9953  alephdom2  10009  alephfp  10030  dfac12lem2  10067  dju1p1e2ALT  10097  cfsmolem  10192  ttukeylem6  10436  ttukeylem7  10437  unsnen  10475  efgmnvl  19689  nogt01o  27660  cutbdaybnd2lim  27789  lesrec  27791  bday1  27806  cuteq1  27809  newbday  27894  negsproplem7  28026  mulsproplem13  28120  mulsproplem14  28121  ltonold  28253  addonbday  28271  bdaypw2n0bndlem  28455  z12bdaylem  28476  hfuni  36366  finxpsuclem  37713  pwfi2f1o  43524  nelsubc3  49546