Theorem onelond 37163

Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. Lemma 1.3 of [Schloeder] p. 1. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)

Hypotheses

Ref	Expression
onelond.1	⊢ (𝜑 → 𝐴 ∈ On)
onelond.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
onelond	⊢ (𝜑 → 𝐵 ∈ On)

Proof of Theorem onelond

Step	Hyp	Ref	Expression
1		onelond.1	. 2 ⊢ (𝜑 → 𝐴 ∈ On)
2		onelond.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		onelon 6413	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → 𝐵 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Oncon0 6388

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 5255  ax-pr 5398

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 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392

This theorem is used by:  nadddilem1  37184  nadddilem2  37185  nadddilem3  37186  nadddilem4  37187