Theorem eloni 4375

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 4373	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2148  Ord word 4362  Oncon0 4363

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368

This theorem is referenced by:  elon2  4376  onelon  4384  onin  4386  onelss  4387  ontr1  4389  onordi  4426  onss  4492  onsuc  4500  onsucb  4502  onsucmin  4506  onsucelsucr  4507  onintonm  4516  ordsucunielexmid  4530  onsucuni2  4563  nnord  4611  tfrlem1  6308  tfrlemisucaccv  6325  tfrlemibfn  6328  tfrlemiubacc  6330  tfrexlem  6334  tfr1onlemsucfn  6340  tfr1onlemsucaccv  6341  tfr1onlembfn  6344  tfr1onlemubacc  6346  tfrcllemsucfn  6353  tfrcllemsucaccv  6354  tfrcllembfn  6357  tfrcllemubacc  6359  sucinc2  6446  phplem4on  6866  ordiso  7034