Theorem eloni 4463

Description: An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
eloni	⊢ (𝐴 ∈ On → Ord 𝐴)

Proof of Theorem eloni

Step	Hyp	Ref	Expression
1		elong 4461	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2200  Ord word 4450  Oncon0 4451

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4454  df-on 4456

This theorem is referenced by:  elon2  4464  onelon  4472  onin  4474  onelss  4475  ontr1  4477  onordi  4514  onss  4582  onsuc  4590  onsucb  4592  onsucmin  4596  onsucelsucr  4597  onintonm  4606  ordsucunielexmid  4620  onsucuni2  4653  nnord  4701  tfrlem1  6444  tfrlemisucaccv  6461  tfrlemibfn  6464  tfrlemiubacc  6466  tfrexlem  6470  tfr1onlemsucfn  6476  tfr1onlemsucaccv  6477  tfr1onlembfn  6480  tfr1onlemubacc  6482  tfrcllemsucfn  6489  tfrcllemsucaccv  6490  tfrcllembfn  6493  tfrcllemubacc  6495  sucinc2  6582  phplem4on  7017  ordiso  7191