Theorem onelss 6235

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 6203	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 6209	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 415	. 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 3938  Ord word 6192  Oncon0 6193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-v 3498  df-in 3945  df-ss 3954  df-uni 4841  df-tr 5175  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197

This theorem is referenced by:  ordunidif  6241  onelssi  6301  ssorduni  7502  suceloni  7530  tfisi  7575  tfrlem9  8023  tfrlem11  8026  oaordex  8186  oaass  8189  odi  8207  omass  8208  oewordri  8220  nnaordex  8266  domtriord  8665  hartogs  9010  card2on  9020  tskwe  9381  infxpenlem  9441  cfub  9673  cfsuc  9681  coflim  9685  hsmexlem2  9851  ondomon  9987  pwcfsdom  10007  inar1  10199  tskord  10204  grudomon  10241  gruina  10242  dfrdg2  33042  poseq  33097  sltres  33171  nosupno  33205  aomclem6  39666  iscard5  39908