Theorem eloni 4515

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 4513	. 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 2209   Ord word 4502   Oncon0 4503

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 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 theorem 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 3931  df-tr 4225  df-iord 4506  df-on 4508

This theorem is referenced by:  elon2  4516  onelon  4524  onin  4526  onelss  4527  ontr1  4529  onordi  4566  onss  4635  onsuc  4643  onsucb  4645  onsucmin  4649  onsucelsucr  4650  onintonm  4659  ordsucunielexmid  4673  onsucuni2  4706  nnord  4754  tfrlem1  6569  tfrlemisucaccv  6586  tfrlemibfn  6589  tfrlemiubacc  6591  tfrexlem  6595  tfr1onlemsucfn  6601  tfr1onlemsucaccv  6602  tfr1onlembfn  6605  tfr1onlemubacc  6607  tfrcllemsucfn  6614  tfrcllemsucaccv  6615  tfrcllembfn  6618  tfrcllemubacc  6620  sucinc2  6709  phplem4on  7159  ordiso  7366