Theorem eloni 4396

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 4394	. 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 2160   Ord word 4383   Oncon0 4384

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3828  df-tr 4120  df-iord 4387  df-on 4389

This theorem is referenced by:  elon2  4397  onelon  4405  onin  4407  onelss  4408  ontr1  4410  onordi  4447  onss  4513  onsuc  4521  onsucb  4523  onsucmin  4527  onsucelsucr  4528  onintonm  4537  ordsucunielexmid  4551  onsucuni2  4584  nnord  4632  tfrlem1  6337  tfrlemisucaccv  6354  tfrlemibfn  6357  tfrlemiubacc  6359  tfrexlem  6363  tfr1onlemsucfn  6369  tfr1onlemsucaccv  6370  tfr1onlembfn  6373  tfr1onlemubacc  6375  tfrcllemsucfn  6382  tfrcllemsucaccv  6383  tfrcllembfn  6386  tfrcllemubacc  6388  sucinc2  6475  phplem4on  6899  ordiso  7069