Theorem onelon 4483

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 4474	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelon 4482	. 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 2201  Ord word 4461  Oncon0 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-in 3205  df-ss 3212  df-uni 3895  df-tr 4189  df-iord 4465  df-on 4467

This theorem is referenced by:  oneli  4527  ssorduni  4587  unon  4611  tfrlemibacc  6497  tfrlemibxssdm  6498  tfrlemibfn  6499  tfrexlem  6505  tfr1onlemsucaccv  6512  tfrcllemsucaccv  6525  sucinc2  6619  oav2  6636  omv2  6638