Theorem eloni 4498

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 4496	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2205  Ord word 4485  Oncon0 4486

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3219  df-ss 3226  df-uni 3917  df-tr 4211  df-iord 4489  df-on 4491

This theorem is referenced by:  elon2  4499  onelon  4507  onin  4509  onelss  4510  ontr1  4512  onordi  4549  onss  4617  onsuc  4625  onsucb  4627  onsucmin  4631  onsucelsucr  4632  onintonm  4641  ordsucunielexmid  4655  onsucuni2  4688  nnord  4736  tfrlem1  6541  tfrlemisucaccv  6558  tfrlemibfn  6561  tfrlemiubacc  6563  tfrexlem  6567  tfr1onlemsucfn  6573  tfr1onlemsucaccv  6574  tfr1onlembfn  6577  tfr1onlemubacc  6579  tfrcllemsucfn  6586  tfrcllemsucaccv  6587  tfrcllembfn  6590  tfrcllemubacc  6592  sucinc2  6681  phplem4on  7124  ordiso  7329