Theorem eloni 4440

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 4438	. 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 2178   Ord word 4427   Oncon0 4428

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by:  elon2  4441  onelon  4449  onin  4451  onelss  4452  ontr1  4454  onordi  4491  onss  4559  onsuc  4567  onsucb  4569  onsucmin  4573  onsucelsucr  4574  onintonm  4583  ordsucunielexmid  4597  onsucuni2  4630  nnord  4678  tfrlem1  6417  tfrlemisucaccv  6434  tfrlemibfn  6437  tfrlemiubacc  6439  tfrexlem  6443  tfr1onlemsucfn  6449  tfr1onlemsucaccv  6450  tfr1onlembfn  6453  tfr1onlemubacc  6455  tfrcllemsucfn  6462  tfrcllemsucaccv  6463  tfrcllembfn  6466  tfrcllemubacc  6468  sucinc2  6555  phplem4on  6990  ordiso  7164