Theorem eloni 4407

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 4405	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2164  Ord word 4394  Oncon0 4395

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400

This theorem is referenced by:  elon2  4408  onelon  4416  onin  4418  onelss  4419  ontr1  4421  onordi  4458  onss  4526  onsuc  4534  onsucb  4536  onsucmin  4540  onsucelsucr  4541  onintonm  4550  ordsucunielexmid  4564  onsucuni2  4597  nnord  4645  tfrlem1  6363  tfrlemisucaccv  6380  tfrlemibfn  6383  tfrlemiubacc  6385  tfrexlem  6389  tfr1onlemsucfn  6395  tfr1onlemsucaccv  6396  tfr1onlembfn  6399  tfr1onlemubacc  6401  tfrcllemsucfn  6408  tfrcllemsucaccv  6409  tfrcllembfn  6412  tfrcllemubacc  6414  sucinc2  6501  phplem4on  6925  ordiso  7097