Theorem wlimss 33823

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.

Step	Hyp	Ref	Expression
1		df-wlim 33807	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2	1	ssrab3 4015	1 ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴

Syntax hints:   ∧ wa 396   = wceq 1539   ≠ wne 2943   ⊆ wss 3887  Predcpred 6201  supcsup 9199  infcinf 9200  WLimcwlim 33805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-wlim 33807

This theorem is referenced by: (None)