Theorem eloni 4427

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 4425	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2177  Ord word 4414  Oncon0 4415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-in 3174  df-ss 3181  df-uni 3854  df-tr 4148  df-iord 4418  df-on 4420

This theorem is referenced by:  elon2  4428  onelon  4436  onin  4438  onelss  4439  ontr1  4441  onordi  4478  onss  4546  onsuc  4554  onsucb  4556  onsucmin  4560  onsucelsucr  4561  onintonm  4570  ordsucunielexmid  4584  onsucuni2  4617  nnord  4665  tfrlem1  6404  tfrlemisucaccv  6421  tfrlemibfn  6424  tfrlemiubacc  6426  tfrexlem  6430  tfr1onlemsucfn  6436  tfr1onlemsucaccv  6437  tfr1onlembfn  6440  tfr1onlemubacc  6442  tfrcllemsucfn  6449  tfrcllemsucaccv  6450  tfrcllembfn  6453  tfrcllemubacc  6455  sucinc2  6542  phplem4on  6976  ordiso  7150