MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onelss Structured version   Visualization version   GIF version

Theorem onelss 5725
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 → (𝐵𝐴𝐵𝐴))

Proof of Theorem onelss
StepHypRef Expression
1 eloni 5692 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 5698 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 450 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 17 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3555  Ord word 5681  Oncon0 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403  df-tr 4713  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686
This theorem is referenced by:  ordunidif  5732  onelssi  5795  ssorduni  6932  suceloni  6960  tfisi  7005  tfrlem9  7426  tfrlem11  7429  oaordex  7583  oaass  7586  odi  7604  omass  7605  oewordri  7617  nnaordex  7663  domtriord  8050  hartogs  8393  card2on  8403  tskwe  8720  infxpenlem  8780  cfub  9015  cfsuc  9023  coflim  9027  hsmexlem2  9193  ondomon  9329  pwcfsdom  9349  inar1  9541  tskord  9546  grudomon  9583  gruina  9584  dfrdg2  31402  poseq  31451  sltres  31518  nobndup  31563  nobnddown  31564  aomclem6  37109
  Copyright terms: Public domain W3C validator