Theorem onelss 6293

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 6261	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 6267	. . 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 2108   ⊆ wss 3883  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5188  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  ordunidif  6299  onelssi  6360  ssorduni  7606  suceloni  7635  tfisi  7680  tfrlem9  8187  tfrlem11  8190  oaordex  8351  oaass  8354  odi  8372  omass  8373  oewordri  8385  nnaordex  8431  domtriord  8859  hartogs  9233  card2on  9243  tskwe  9639  infxpenlem  9700  cfub  9936  cfsuc  9944  coflim  9948  hsmexlem2  10114  ondomon  10250  pwcfsdom  10270  inar1  10462  tskord  10467  grudomon  10504  gruina  10505  dfrdg2  33677  poseq  33729  sltres  33792  nosupno  33833  nosupbday  33835  noinfno  33848  oldssmade  33987  madebday  34007  aomclem6  40800  iscard5  41039