Theorem onelss 6360

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 6328	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 6334	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 412	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3902  Ord word 6317  Oncon0 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3443  df-ss 3919  df-uni 4865  df-tr 5207  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322

This theorem is referenced by:  ordunidif  6368  onelssi  6434  ssorduni  7726  tfisi  7803  poseq  8102  tfrlem9  8318  tfrlem11  8321  oaordex  8487  oaass  8490  odi  8508  omass  8509  oewordri  8522  nnaordex  8568  domtriord  9055  hartogs  9453  card2on  9463  tskwe  9866  infxpenlem  9927  cfub  10163  cfsuc  10171  coflim  10175  hsmexlem2  10341  ondomon  10477  pwcfsdom  10498  inar1  10690  tskord  10695  grudomon  10732  gruina  10733  sltres  27634  nosupno  27675  nosupbday  27677  noinfno  27690  oldssmade  27859  madebday  27882  mulsproplem13  28110  mulsproplem14  28111  dfrdg2  35968  aomclem6  43337  nnoeomeqom  43590  naddgeoa  43672  naddwordnexlem1  43675  naddwordnexlem4  43679  iscard5  43813