Theorem eloni 4360

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 4358	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 175	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2141  Ord word 4347  Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  elon2  4361  onelon  4369  onin  4371  onelss  4372  ontr1  4374  onordi  4411  onss  4477  suceloni  4485  sucelon  4487  onsucmin  4491  onsucelsucr  4492  onintonm  4501  ordsucunielexmid  4515  onsucuni2  4548  nnord  4596  tfrlem1  6287  tfrlemisucaccv  6304  tfrlemibfn  6307  tfrlemiubacc  6309  tfrexlem  6313  tfr1onlemsucfn  6319  tfr1onlemsucaccv  6320  tfr1onlembfn  6323  tfr1onlemubacc  6325  tfrcllemsucfn  6332  tfrcllemsucaccv  6333  tfrcllembfn  6336  tfrcllemubacc  6338  sucinc2  6425  phplem4on  6845  ordiso  7013