Theorem eloni 4376

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 4374	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2148  Ord word 4363  Oncon0 4364

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 2740  df-in 3136  df-ss 3143  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369

This theorem is referenced by:  elon2  4377  onelon  4385  onin  4387  onelss  4388  ontr1  4390  onordi  4427  onss  4493  onsuc  4501  onsucb  4503  onsucmin  4507  onsucelsucr  4508  onintonm  4517  ordsucunielexmid  4531  onsucuni2  4564  nnord  4612  tfrlem1  6309  tfrlemisucaccv  6326  tfrlemibfn  6329  tfrlemiubacc  6331  tfrexlem  6335  tfr1onlemsucfn  6341  tfr1onlemsucaccv  6342  tfr1onlembfn  6345  tfr1onlemubacc  6347  tfrcllemsucfn  6354  tfrcllemsucaccv  6355  tfrcllembfn  6358  tfrcllemubacc  6360  sucinc2  6447  phplem4on  6867  ordiso  7035