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 32643
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 32627 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
21ssrab3 3947 1 WLim(𝑅, 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 387   = wceq 1507  wne 2967  wss 3829  Predcpred 5985  supcsup 8699  infcinf 8700  WLimcwlim 32625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rab 3097  df-in 3836  df-ss 3843  df-wlim 32627
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator