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

Theorem limelon 5747
Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
limelon ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Proof of Theorem limelon
StepHypRef Expression
1 limord 5743 . . 3 (Lim 𝐴 → Ord 𝐴)
2 elong 5690 . . 3 (𝐴𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2syl5ibr 236 . 2 (𝐴𝐵 → (Lim 𝐴𝐴 ∈ On))
43imp 445 1 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Ord word 5681  Oncon0 5682  Lim wlim 5683
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-3an 1038  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  df-lim 5687
This theorem is referenced by:  onzsl  6993  limuni3  6999  tfindsg2  7008  dfom2  7014  rdglim  7467  oalim  7557  omlim  7558  oelim  7559  oalimcl  7585  oaass  7586  omlimcl  7603  odi  7604  omass  7605  oen0  7611  oewordri  7617  oelim2  7620  oelimcl  7625  omabs  7672  r1lim  8579  alephordi  8841  cflm  9016  alephsing  9042  pwcfsdom  9349  winafp  9463  r1limwun  9502
  Copyright terms: Public domain W3C validator