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Theorem wlimss 30856
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 30836 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2 ssrab2 3649 . 2 {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} ⊆ 𝐴
31, 2eqsstri 3597 1 WLim(𝑅, 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wne 2779  {crab 2899  wss 3539  Predcpred 5582  supcsup 8207  infcinf 8208  WLimcwlim 30832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-in 3546  df-ss 3553  df-wlim 30836
This theorem is referenced by: (None)
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