Theorem onelssd 36693

Description: An element of an ordinal number is a subset of the number. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)

Hypotheses

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

Assertion

Ref	Expression
onelssd	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Proof of Theorem onelssd

Step	Hyp	Ref	Expression
1		onelssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ On)
2		onelssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		onelss 6403	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 2, 3	sylc 66	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2143   ⊆ wss 3905  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

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3922  df-uni 4873  df-tr 5219  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364

This theorem is referenced by:  nadddilem3  36714