Theorem onelond 37072

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 6390	. 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 2144  Oncon0 6365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368  df-on 6369

This theorem is used by:  nadddilem1  37093  nadddilem2  37094  nadddilem3  37095  nadddilem4  37096