Theorem onelssi 6481

Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onelssi	⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

Proof of Theorem onelssi

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		onelss 6407	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ⊆ wss 3906  Oncon0 6364

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-uni 4875  df-tr 5221  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368

This theorem is used by:  onelini  6484  oneluni  6485  oawordeulem  8545  cardsdomelir  9975  carddom2  9979  cardaleph  10089  alephsing  10275  domtriomlem  10441  axdc3lem  10449  inar1  10779  nodense  27911