Theorem oneli 6499

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 6408	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
3	1, 2	mpan 703	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Oncon0 6383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-tr 5217  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6386  df-on 6387

This theorem is used by:  onssneli  6501  oawordeulem  8593  rankuni  9909  tcrank  9931  cardne  10076  cardval2  10102  alephsuc2  10189  cfsmolem  10378  cfcof  10382  alephreg  10697  pwcfsdom  10698  tskcard  10896  lrcut  28323  addonbday  28698  j0ltaleph  36021  jltalephlem  36038  jltaleph  36039  fnlfvk3n0  36041  onvf1odlem4  36047  onsucconni  37416