Theorem eloni 4465

Description: An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
eloni	|- ( A e. On -> Ord A )

Proof of Theorem eloni

Step	Hyp	Ref	Expression
1		elong 4463	. 2 |- ( A e. On -> ( A e. On <-> Ord A ) )
2	1	ibi 176	1 |- ( A e. On -> Ord A )

Syntax hints:   -> wi 4   e. wcel 2200   Ord word 4452   Oncon0 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458

This theorem is referenced by:  elon2  4466  onelon  4474  onin  4476  onelss  4477  ontr1  4479  onordi  4516  onss  4584  onsuc  4592  onsucb  4594  onsucmin  4598  onsucelsucr  4599  onintonm  4608  ordsucunielexmid  4622  onsucuni2  4655  nnord  4703  tfrlem1  6452  tfrlemisucaccv  6469  tfrlemibfn  6472  tfrlemiubacc  6474  tfrexlem  6478  tfr1onlemsucfn  6484  tfr1onlemsucaccv  6485  tfr1onlembfn  6488  tfr1onlemubacc  6490  tfrcllemsucfn  6497  tfrcllemsucaccv  6498  tfrcllembfn  6501  tfrcllemubacc  6503  sucinc2  6590  phplem4on  7025  ordiso  7199