Theorem onelond 36780

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 6382	. 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 2145  Oncon0 6357

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 2732  ax-sep 5251  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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361

This theorem is used by:  nadddilem1  36801  nadddilem2  36802  nadddilem3  36803  nadddilem4  36804