Theorem wlimss 31778

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 31758	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
2		ssrab2 3687	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} ⊆ 𝐴
3	1, 2	eqsstri 3635	1 ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴

Syntax hints:   ∧ wa 384   = wceq 1483   ≠ wne 2794  {crab 2916   ⊆ wss 3574  Predcpred 5679  supcsup 8346  infcinf 8347  WLimcwlim 31754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588  df-wlim 31758

This theorem is referenced by: (None)