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Definition df-wlim 32095
Description: Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
Assertion
Ref Expression
df-wlim WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Detailed syntax breakdown of Definition df-wlim
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2cwlim 32093 . 2 class WLim(𝑅, 𝐴)
4 vx . . . . . 6 setvar 𝑥
54cv 1630 . . . . 5 class 𝑥
61, 1, 2cinf 8503 . . . . 5 class inf(𝐴, 𝐴, 𝑅)
75, 6wne 2943 . . . 4 wff 𝑥 ≠ inf(𝐴, 𝐴, 𝑅)
81, 2, 5cpred 5822 . . . . . 6 class Pred(𝑅, 𝐴, 𝑥)
98, 1, 2csup 8502 . . . . 5 class sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅)
105, 9wceq 1631 . . . 4 wff 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅)
117, 10wa 382 . . 3 wff (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))
1211, 4, 1crab 3065 . 2 class {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
133, 12wceq 1631 1 wff WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
Colors of variables: wff setvar class
This definition is referenced by:  wlimeq12  32101  nfwlim  32104  elwlim  32105  wlimss  32111
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