Theorem eloni 4466

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 4464	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2200  Ord word 4453  Oncon0 4454

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 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  elon2  4467  onelon  4475  onin  4477  onelss  4478  ontr1  4480  onordi  4517  onss  4585  onsuc  4593  onsucb  4595  onsucmin  4599  onsucelsucr  4600  onintonm  4609  ordsucunielexmid  4623  onsucuni2  4656  nnord  4704  tfrlem1  6460  tfrlemisucaccv  6477  tfrlemibfn  6480  tfrlemiubacc  6482  tfrexlem  6486  tfr1onlemsucfn  6492  tfr1onlemsucaccv  6493  tfr1onlembfn  6496  tfr1onlemubacc  6498  tfrcllemsucfn  6505  tfrcllemsucaccv  6506  tfrcllembfn  6509  tfrcllemubacc  6511  sucinc2  6600  phplem4on  7037  ordiso  7211