Theorem eloni 4501

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 4499	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2205  Ord word 4488  Oncon0 4489

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 3220  df-ss 3227  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494

This theorem is referenced by:  elon2  4502  onelon  4510  onin  4512  onelss  4513  ontr1  4515  onordi  4552  onss  4620  onsuc  4628  onsucb  4630  onsucmin  4634  onsucelsucr  4635  onintonm  4644  ordsucunielexmid  4658  onsucuni2  4691  nnord  4739  tfrlem1  6552  tfrlemisucaccv  6569  tfrlemibfn  6572  tfrlemiubacc  6574  tfrexlem  6578  tfr1onlemsucfn  6584  tfr1onlemsucaccv  6585  tfr1onlembfn  6588  tfr1onlemubacc  6590  tfrcllemsucfn  6597  tfrcllemsucaccv  6598  tfrcllembfn  6601  tfrcllemubacc  6603  sucinc2  6692  phplem4on  7135  ordiso  7340