Theorem eloni 4478

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 4476	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2202  Ord word 4465  Oncon0 4466

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471

This theorem is referenced by:  elon2  4479  onelon  4487  onin  4489  onelss  4490  ontr1  4492  onordi  4529  onss  4597  onsuc  4605  onsucb  4607  onsucmin  4611  onsucelsucr  4612  onintonm  4621  ordsucunielexmid  4635  onsucuni2  4668  nnord  4716  tfrlem1  6517  tfrlemisucaccv  6534  tfrlemibfn  6537  tfrlemiubacc  6539  tfrexlem  6543  tfr1onlemsucfn  6549  tfr1onlemsucaccv  6550  tfr1onlembfn  6553  tfr1onlemubacc  6555  tfrcllemsucfn  6562  tfrcllemsucaccv  6563  tfrcllembfn  6566  tfrcllemubacc  6568  sucinc2  6657  phplem4on  7097  ordiso  7278