Theorem eloni 4410

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 4408	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	ibi 176	1 ⊢ (𝐴 ∈ On → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2167  Ord word 4397  Oncon0 4398

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 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by:  elon2  4411  onelon  4419  onin  4421  onelss  4422  ontr1  4424  onordi  4461  onss  4529  onsuc  4537  onsucb  4539  onsucmin  4543  onsucelsucr  4544  onintonm  4553  ordsucunielexmid  4567  onsucuni2  4600  nnord  4648  tfrlem1  6366  tfrlemisucaccv  6383  tfrlemibfn  6386  tfrlemiubacc  6388  tfrexlem  6392  tfr1onlemsucfn  6398  tfr1onlemsucaccv  6399  tfr1onlembfn  6402  tfr1onlemubacc  6404  tfrcllemsucfn  6411  tfrcllemsucaccv  6412  tfrcllembfn  6415  tfrcllemubacc  6417  sucinc2  6504  phplem4on  6928  ordiso  7102