Theorem onordi 6438

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 6335	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ord 𝐴

Syntax hints:   ∈ wcel 2114  Ord word 6324  Oncon0 6325

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-ss 3920  df-uni 4866  df-tr 5208  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  onirri  6439  onsucssi  7793  ord1eln01  8433  ord2eln012  8434  oawordeulem  8491  omopthi  8599  en2  9192  en3  9193  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  19655  nogt01o  27676  cutbdaybnd2lim  27805  lesrec  27807  bday1  27822  cuteq1  27825  newbday  27910  negsproplem7  28042  mulsproplem13  28136  mulsproplem14  28137  ltonold  28269  addonbday  28287  bdaypw2n0bndlem  28471  z12bdaylem  28492  hfuni  36400  finxpsuclem  37652  pwfi2f1o  43453  nelsubc3  49430