Theorem wlimss 35824

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 35808	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2	1	ssrab3 4048	1 ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1540   ≠ wne 2926   ⊆ wss 3917  Predcpred 6276  supcsup 9398  infcinf 9399  WLimcwlim 35806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-ss 3934  df-wlim 35808

This theorem is referenced by: (None)