Theorem onordi 6453

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 6350	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ord 𝐴

Syntax hints:   ∈ wcel 2141  Ord word 6339  Oncon0 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-ss 3921  df-uni 4865  df-tr 5207  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6343  df-on 6344

This theorem is referenced by:  onirri  6454  onsucssi  7815  ord1eln01  8458  ord2eln012  8459  oawordeulem  8516  omopthi  8624  en2  9218  en3  9219  ssttrcl  9665  ttrcltr  9666  dmttrcl  9671  ttrclselem2  9676  bndrank  9794  rankprb  9804  rankuniss  9819  rankelun  9825  rankelpr  9826  rankelop  9827  rankmapu  9831  rankxplim3  9834  rankxpsuc  9835  cardlim  9925  carduni  9934  dfac8b  9982  alephdom2  10038  alephfp  10059  dfac12lem2  10096  dju1p1e2ALT  10126  cfsmolem  10222  ttukeylem6  10466  ttukeylem7  10467  unsnen  10505  efgmnvl  19735  nogt01o  27735  cutbdaybnd2lim  27865  lesrec  27867  bday1  27882  cuteq1  27885  newbday  27970  negsproplem7  28102  mulsproplem13  28196  mulsproplem14  28197  ltonold  28329  addonbday  28347  bdaypw2n0bndlem  28531  z12bdaylem  28552  hfuni  36487  finxpsuclem  37844  pwfi2f1o  43626  nelsubc3  49645