Theorem onelssd 37074

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 6408	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 2, 3	sylc 66	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ⊆ wss 3896  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

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3897  df-uni 4867  df-tr 5212  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368  df-on 6369

This theorem is used by:  nadddilem3  37095