Theorem onordi 6431

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 6328	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ord 𝐴

Syntax hints:   ∈ wcel 2114  Ord word 6317  Oncon0 6318

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 3432  df-ss 3907  df-uni 4852  df-tr 5194  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322

This theorem is referenced by:  onirri  6432  onsucssi  7786  ord1eln01  8425  ord2eln012  8426  oawordeulem  8483  omopthi  8591  en2  9184  en3  9185  ssttrcl  9630  ttrcltr  9631  dmttrcl  9636  ttrclselem2  9641  bndrank  9759  rankprb  9769  rankuniss  9784  rankelun  9790  rankelpr  9791  rankelop  9792  rankmapu  9796  rankxplim3  9799  rankxpsuc  9800  cardlim  9890  carduni  9899  dfac8b  9947  alephdom2  10003  alephfp  10024  dfac12lem2  10061  dju1p1e2ALT  10091  cfsmolem  10186  ttukeylem6  10430  ttukeylem7  10431  unsnen  10469  efgmnvl  19683  nogt01o  27677  cutbdaybnd2lim  27806  lesrec  27808  bday1  27823  cuteq1  27826  newbday  27911  negsproplem7  28043  mulsproplem13  28137  mulsproplem14  28138  ltonold  28270  addonbday  28288  bdaypw2n0bndlem  28472  z12bdaylem  28493  hfuni  36385  finxpsuclem  37730  pwfi2f1o  43545  nelsubc3  49561