Theorem onelss 6413

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 6381	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 6387	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 411	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3944  Ord word 6370  Oncon0 6371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-v 3463  df-ss 3961  df-uni 4910  df-tr 5267  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375

This theorem is referenced by:  ordunidif  6420  onelssi  6486  ssorduni  7782  sucexeloniOLD  7814  suceloniOLD  7816  tfisi  7864  poseq  8163  tfrlem9  8406  tfrlem11  8409  oaordex  8579  oaass  8582  odi  8600  omass  8601  oewordri  8613  nnaordex  8659  domtriord  9148  hartogs  9569  card2on  9579  tskwe  9975  infxpenlem  10038  cfub  10274  cfsuc  10282  coflim  10286  hsmexlem2  10452  ondomon  10588  pwcfsdom  10608  inar1  10800  tskord  10805  grudomon  10842  gruina  10843  sltres  27641  nosupno  27682  nosupbday  27684  noinfno  27697  oldssmade  27850  madebday  27872  mulsproplem13  28078  mulsproplem14  28079  dfrdg2  35522  aomclem6  42625  nnoeomeqom  42883  naddgeoa  42966  naddwordnexlem1  42969  naddwordnexlem4  42973  iscard5  43108