Theorem onelord 44094

Description: Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6386 and eloni 6371. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
onelord	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Proof of Theorem onelord

Step	Hyp	Ref	Expression
1		onelon 6386	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		eloni 6371	. 2 ⊢ (𝐵 ∈ On → Ord 𝐵)
3	1, 2	syl 18	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  Ord word 6360  Oncon0 6361

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365

This theorem is used by: (None)