Theorem onelssi 6498

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 6425	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107   ⊆ wss 3950  Oncon0 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-ss 3967  df-uni 4907  df-tr 5259  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387

This theorem is referenced by:  onelini  6501  oneluni  6502  oawordeulem  8593  cardsdomelir  10014  carddom2  10018  cardaleph  10130  alephsing  10317  domtriomlem  10483  axdc3lem  10491  inar1  10816  nodense  27738