Theorem eloni 4466

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 4464	. 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 4453   Oncon0 4454

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 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  elon2  4467  onelon  4475  onin  4477  onelss  4478  ontr1  4480  onordi  4517  onss  4585  onsuc  4593  onsucb  4595  onsucmin  4599  onsucelsucr  4600  onintonm  4609  ordsucunielexmid  4623  onsucuni2  4656  nnord  4704  tfrlem1  6454  tfrlemisucaccv  6471  tfrlemibfn  6474  tfrlemiubacc  6476  tfrexlem  6480  tfr1onlemsucfn  6486  tfr1onlemsucaccv  6487  tfr1onlembfn  6490  tfr1onlemubacc  6492  tfrcllemsucfn  6499  tfrcllemsucaccv  6500  tfrcllembfn  6503  tfrcllemubacc  6505  sucinc2  6592  phplem4on  7029  ordiso  7203