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.

Step	Hyp	Ref	Expression
1		df-wlim 32627	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2	1	ssrab3 3947	1 ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴

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)