Theorem onelss 5908

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 5875	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 5881	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 397	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2145   ⊆ wss 3723  Ord word 5864  Oncon0 5865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-uni 4576  df-tr 4888  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868  df-on 5869

This theorem is referenced by:  ordunidif  5915  onelssi  5978  ssorduni  7136  suceloni  7164  tfisi  7209  tfrlem9  7638  tfrlem11  7641  oaordex  7796  oaass  7799  odi  7817  omass  7818  oewordri  7830  nnaordex  7876  domtriord  8266  hartogs  8609  card2on  8619  tskwe  8980  infxpenlem  9040  cfub  9277  cfsuc  9285  coflim  9289  hsmexlem2  9455  ondomon  9591  pwcfsdom  9611  inar1  9803  tskord  9808  grudomon  9845  gruina  9846  dfrdg2  32037  poseq  32090  sltres  32152  nosupno  32186  aomclem6  38153