Theorem eloni 4377

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 4375	. 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 2148   Ord word 4364   Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  elon2  4378  onelon  4386  onin  4388  onelss  4389  ontr1  4391  onordi  4428  onss  4494  onsuc  4502  onsucb  4504  onsucmin  4508  onsucelsucr  4509  onintonm  4518  ordsucunielexmid  4532  onsucuni2  4565  nnord  4613  tfrlem1  6311  tfrlemisucaccv  6328  tfrlemibfn  6331  tfrlemiubacc  6333  tfrexlem  6337  tfr1onlemsucfn  6343  tfr1onlemsucaccv  6344  tfr1onlembfn  6347  tfr1onlemubacc  6349  tfrcllemsucfn  6356  tfrcllemsucaccv  6357  tfrcllembfn  6360  tfrcllemubacc  6362  sucinc2  6449  phplem4on  6869  ordiso  7037