Theorem eloni 4520

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 4518	. 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 4507   Oncon0 4508

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 4230  df-iord 4511  df-on 4513

This theorem is used by:  elon2  4521  onelon  4529  onin  4531  onelss  4532  ontr1  4534  onordi  4571  onss  4640  onsuc  4648  onsucb  4650  onsucmin  4654  onsucelsucr  4655  onintonm  4664  ordsucunielexmid  4678  onsucuni2  4711  nnord  4759  tfrlem1  6579  tfrlemisucaccv  6596  tfrlemibfn  6599  tfrlemiubacc  6601  tfrexlem  6605  tfr1onlemsucfn  6611  tfr1onlemsucaccv  6612  tfr1onlembfn  6615  tfr1onlemubacc  6617  tfrcllemsucfn  6624  tfrcllemsucaccv  6625  tfrcllembfn  6628  tfrcllemubacc  6630  sucinc2  6719  phplem4on  7169  ordiso  7376