Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlimss Structured version   Visualization version   GIF version

Theorem wlimss 35811
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 35795 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
21ssrab3 4092 1 WLim(𝑅, 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wne 2938  wss 3963  Predcpred 6322  supcsup 9478  infcinf 9479  WLimcwlim 35793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-ss 3980  df-wlim 35795
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator