Theorem eloni 4411

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 4409	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2167  Ord word 4398  Oncon0 4399

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404

This theorem is referenced by:  elon2  4412  onelon  4420  onin  4422  onelss  4423  ontr1  4425  onordi  4462  onss  4530  onsuc  4538  onsucb  4540  onsucmin  4544  onsucelsucr  4545  onintonm  4554  ordsucunielexmid  4568  onsucuni2  4601  nnord  4649  tfrlem1  6375  tfrlemisucaccv  6392  tfrlemibfn  6395  tfrlemiubacc  6397  tfrexlem  6401  tfr1onlemsucfn  6407  tfr1onlemsucaccv  6408  tfr1onlembfn  6411  tfr1onlemubacc  6413  tfrcllemsucfn  6420  tfrcllemsucaccv  6421  tfrcllembfn  6424  tfrcllemubacc  6426  sucinc2  6513  phplem4on  6937  ordiso  7111