Theorem onelssi 6479

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 6405	. 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 2145   ⊆ wss 3899  Oncon0 6362

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366

This theorem is used by:  onelini  6482  oneluni  6483  oawordeulem  8562  cardsdomelir  10054  carddom2  10058  cardaleph  10168  alephsing  10354  domtriomlem  10520  axdc3lem  10528  inar1  10860  nodense  28049