Theorem eloni 4467

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 4465	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2200  Ord word 4454  Oncon0 4455

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 4458  df-on 4460

This theorem is referenced by:  elon2  4468  onelon  4476  onin  4478  onelss  4479  ontr1  4481  onordi  4518  onss  4586  onsuc  4594  onsucb  4596  onsucmin  4600  onsucelsucr  4601  onintonm  4610  ordsucunielexmid  4624  onsucuni2  4657  nnord  4705  tfrlem1  6465  tfrlemisucaccv  6482  tfrlemibfn  6485  tfrlemiubacc  6487  tfrexlem  6491  tfr1onlemsucfn  6497  tfr1onlemsucaccv  6498  tfr1onlembfn  6501  tfr1onlemubacc  6503  tfrcllemsucfn  6510  tfrcllemsucaccv  6511  tfrcllembfn  6514  tfrcllemubacc  6516  sucinc2  6605  phplem4on  7042  ordiso  7219