Theorem eloni 4522

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 4520	. 2 |- ( A e. On -> ( A e. On <-> Ord A ) )
2	1	ibi 176	1 |- ( A e. On -> Ord A )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   Ord word 4509   Oncon0 4510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4232  df-iord 4513  df-on 4515

This theorem is used by:  elon2  4523  onelon  4531  onin  4533  onelss  4534  ontr1  4536  onordi  4573  onss  4642  onsuc  4650  onsucb  4652  onsucmin  4656  onsucelsucr  4657  onintonm  4666  ordsucunielexmid  4680  onsucuni2  4713  nnord  4761  tfrlem1  6583  tfrlemisucaccv  6600  tfrlemibfn  6603  tfrlemiubacc  6605  tfrexlem  6609  tfr1onlemsucfn  6615  tfr1onlemsucaccv  6616  tfr1onlembfn  6619  tfr1onlemubacc  6621  tfrcllemsucfn  6628  tfrcllemsucaccv  6629  tfrcllembfn  6632  tfrcllemubacc  6634  sucinc2  6723  phplem4on  7173  ordiso  7382