Theorem oneli 6359

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 6276	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
3	1, 2	mpan 686	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2108  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  onssneli  6361  oawordeulem  8347  rankuni  9552  tcrank  9573  cardne  9654  cardval2  9680  alephsuc2  9767  cfsmolem  9957  cfcof  9961  alephreg  10269  pwcfsdom  10270  tskcard  10468  lrcut  34010  onsucconni  34553