Theorem onelord 43667

Description: Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6337 and eloni 6322. (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 6337	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		eloni 6322	. 2 ⊢ (𝐵 ∈ On → Ord 𝐵)
3	1, 2	syl 17	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Ord word 6311  Oncon0 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6315  df-on 6316

This theorem is referenced by: (None)