Theorem eloni 4470

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 4468	. 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 4457   Oncon0 4458

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 2802  df-in 3204  df-ss 3211  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463

This theorem is referenced by:  elon2  4471  onelon  4479  onin  4481  onelss  4482  ontr1  4484  onordi  4521  onss  4589  onsuc  4597  onsucb  4599  onsucmin  4603  onsucelsucr  4604  onintonm  4613  ordsucunielexmid  4627  onsucuni2  4660  nnord  4708  tfrlem1  6469  tfrlemisucaccv  6486  tfrlemibfn  6489  tfrlemiubacc  6491  tfrexlem  6495  tfr1onlemsucfn  6501  tfr1onlemsucaccv  6502  tfr1onlembfn  6505  tfr1onlemubacc  6507  tfrcllemsucfn  6514  tfrcllemsucaccv  6515  tfrcllembfn  6518  tfrcllemubacc  6520  sucinc2  6609  phplem4on  7049  ordiso  7226