Theorem eloni 4518

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 4516	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2209  Ord word 4505  Oncon0 4506

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 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 theorem 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 3934  df-tr 4228  df-iord 4509  df-on 4511

This theorem is referenced by:  elon2  4519  onelon  4527  onin  4529  onelss  4530  ontr1  4532  onordi  4569  onss  4638  onsuc  4646  onsucb  4648  onsucmin  4652  onsucelsucr  4653  onintonm  4662  ordsucunielexmid  4676  onsucuni2  4709  nnord  4757  tfrlem1  6573  tfrlemisucaccv  6590  tfrlemibfn  6593  tfrlemiubacc  6595  tfrexlem  6599  tfr1onlemsucfn  6605  tfr1onlemsucaccv  6606  tfr1onlembfn  6609  tfr1onlemubacc  6611  tfrcllemsucfn  6618  tfrcllemsucaccv  6619  tfrcllembfn  6622  tfrcllemubacc  6624  sucinc2  6713  phplem4on  7163  ordiso  7370