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Theorem limelon 5705
 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 5701 . . 3 (Lim 𝐴 → Ord 𝐴)
2 elong 5648 . . 3 (𝐴𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2syl5ibr 235 . 2 (𝐴𝐵 → (Lim 𝐴𝐴 ∈ On))
43imp 444 1 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Ord word 5639  Oncon0 5640  Lim wlim 5641 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645 This theorem is referenced by:  onzsl  6938  limuni3  6944  tfindsg2  6953  dfom2  6959  rdglim  7409  oalim  7499  omlim  7500  oelim  7501  oalimcl  7527  oaass  7528  omlimcl  7545  odi  7546  omass  7547  oen0  7553  oewordri  7559  oelim2  7562  oelimcl  7567  omabs  7614  r1lim  8518  alephordi  8780  cflm  8955  alephsing  8981  pwcfsdom  9284  winafp  9398  r1limwun  9437
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