Theorem eloni 4353

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 4351	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 175	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2136  Ord word 4340  Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  elon2  4354  onelon  4362  onin  4364  onelss  4365  ontr1  4367  onordi  4404  onss  4470  suceloni  4478  sucelon  4480  onsucmin  4484  onsucelsucr  4485  onintonm  4494  ordsucunielexmid  4508  onsucuni2  4541  nnord  4589  tfrlem1  6276  tfrlemisucaccv  6293  tfrlemibfn  6296  tfrlemiubacc  6298  tfrexlem  6302  tfr1onlemsucfn  6308  tfr1onlemsucaccv  6309  tfr1onlembfn  6312  tfr1onlemubacc  6314  tfrcllemsucfn  6321  tfrcllemsucaccv  6322  tfrcllembfn  6325  tfrcllemubacc  6327  sucinc2  6414  phplem4on  6833  ordiso  7001