Theorem onelond 36691

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 6385	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → 𝐵 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2143  Oncon0 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364

This theorem is referenced by:  nadddilem1  36712  nadddilem2  36713  nadddilem3  36714  nadddilem4  36715