Theorem eloni 4422

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 4420	. 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 2176   Ord word 4409   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  elon2  4423  onelon  4431  onin  4433  onelss  4434  ontr1  4436  onordi  4473  onss  4541  onsuc  4549  onsucb  4551  onsucmin  4555  onsucelsucr  4556  onintonm  4565  ordsucunielexmid  4579  onsucuni2  4612  nnord  4660  tfrlem1  6394  tfrlemisucaccv  6411  tfrlemibfn  6414  tfrlemiubacc  6416  tfrexlem  6420  tfr1onlemsucfn  6426  tfr1onlemsucaccv  6427  tfr1onlembfn  6430  tfr1onlemubacc  6432  tfrcllemsucfn  6439  tfrcllemsucaccv  6440  tfrcllembfn  6443  tfrcllemubacc  6445  sucinc2  6532  phplem4on  6964  ordiso  7138