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Definition df-ilim 4078
Description: Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes 𝐴 ≠ ∅ to ∅ ∈ 𝐴 (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4079 instead for naming consistency with set.mm. (New usage is discouraged.)
Assertion
Ref Expression
df-ilim (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))

Detailed syntax breakdown of Definition df-ilim
StepHypRef Expression
1 cA . . 3 class 𝐴
21wlim 4073 . 2 wff Lim 𝐴
31word 4071 . . 3 wff Ord 𝐴
4 c0 3221 . . . 4 class
54, 1wcel 1393 . . 3 wff ∅ ∈ 𝐴
61cuni 3577 . . . 4 class 𝐴
71, 6wceq 1243 . . 3 wff 𝐴 = 𝐴
83, 5, 7w3a 885 . 2 wff (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴)
92, 8wb 98 1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
Colors of variables: wff set class
This definition is referenced by:  dflim2  4079
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