Theorem oneli 6478

Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
oneli	⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)

Proof of Theorem oneli

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		onelon 6387	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
3	1, 2	mpan 702	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2143  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  onssneli  6480  oawordeulem  8540  rankuni  9836  tcrank  9857  cardne  9952  cardval2  9978  alephsuc2  10065  cfsmolem  10255  cfcof  10259  alephreg  10568  pwcfsdom  10569  tskcard  10767  lrcut  28078  addonbday  28453  onvf1odlem4  35571  onsucconni  36929