Theorem eloni 4495

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 4493	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2203  Ord word 4482  Oncon0 4483

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-in 3216  df-ss 3223  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488

This theorem is referenced by:  elon2  4496  onelon  4504  onin  4506  onelss  4507  ontr1  4509  onordi  4546  onss  4614  onsuc  4622  onsucb  4624  onsucmin  4628  onsucelsucr  4629  onintonm  4638  ordsucunielexmid  4652  onsucuni2  4685  nnord  4733  tfrlem1  6538  tfrlemisucaccv  6555  tfrlemibfn  6558  tfrlemiubacc  6560  tfrexlem  6564  tfr1onlemsucfn  6570  tfr1onlemsucaccv  6571  tfr1onlembfn  6574  tfr1onlemubacc  6576  tfrcllemsucfn  6583  tfrcllemsucaccv  6584  tfrcllembfn  6587  tfrcllemubacc  6589  sucinc2  6678  phplem4on  7121  ordiso  7326