Theorem onelss 5983

Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
onelss	⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem onelss

Step	Hyp	Ref	Expression
1		eloni 5951	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 5957	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 402	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2157   ⊆ wss 3769  Ord word 5940  Oncon0 5941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-uni 4629  df-tr 4946  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945

This theorem is referenced by:  ordunidif  5989  onelssi  6049  ssorduni  7219  suceloni  7247  tfisi  7292  tfrlem9  7720  tfrlem11  7723  oaordex  7878  oaass  7881  odi  7899  omass  7900  oewordri  7912  nnaordex  7958  domtriord  8348  hartogs  8691  card2on  8701  tskwe  9062  infxpenlem  9122  cfub  9359  cfsuc  9367  coflim  9371  hsmexlem2  9537  ondomon  9673  pwcfsdom  9693  inar1  9885  tskord  9890  grudomon  9927  gruina  9928  dfrdg2  32213  poseq  32266  sltres  32328  nosupno  32362  aomclem6  38414