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Theorem wlimss 31022
 Description: The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
Assertion
Ref Expression
wlimss WLim(𝑅, 𝐴) ⊆ 𝐴

Proof of Theorem wlimss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-wlim 31002 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2 ssrab2 3650 . 2 {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} ⊆ 𝐴
31, 2eqsstri 3598 1 WLim(𝑅, 𝐴) ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ≠ wne 2780  {crab 2900   ⊆ wss 3540  Predcpred 5596  supcsup 8229  infcinf 8230  WLimcwlim 30998 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-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554  df-wlim 31002 This theorem is referenced by: (None)
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