Theorem onelon 4452

Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
onelon	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)

Proof of Theorem onelon

Step	Hyp	Ref	Expression
1		eloni 4443	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelon 4451	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
3	1, 2	sylan 283	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2180  Ord word 4430  Oncon0 4431

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-in 3183  df-ss 3190  df-uni 3868  df-tr 4162  df-iord 4434  df-on 4436

This theorem is referenced by:  oneli  4496  ssorduni  4556  unon  4580  tfrlemibacc  6442  tfrlemibxssdm  6443  tfrlemibfn  6444  tfrexlem  6450  tfr1onlemsucaccv  6457  tfrcllemsucaccv  6470  sucinc2  6562  oav2  6579  omv2  6581