Theorem wlimss 34801

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 34785	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2	1	ssrab3 4081	1 ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴

Syntax hints:   ∧ wa 397   = wceq 1542   ≠ wne 2941   ⊆ wss 3949  Predcpred 6300  supcsup 9435  infcinf 9436  WLimcwlim 34783

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-wlim 34785

This theorem is referenced by: (None)