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Mirrors > Home > MPE Home > Th. List > df-lim | Structured version Visualization version GIF version |
Description: Define the limit ordinal predicate, which is true for a nonempty ordinal that 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. See dflim2 5698, dflim3 6939, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.) |
Ref | Expression |
---|---|
df-lim | ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | wlim 5641 | . 2 wff Lim 𝐴 |
3 | 1 | word 5639 | . . 3 wff Ord 𝐴 |
4 | c0 3874 | . . . 4 class ∅ | |
5 | 1, 4 | wne 2780 | . . 3 wff 𝐴 ≠ ∅ |
6 | 1 | cuni 4372 | . . . 4 class ∪ 𝐴 |
7 | 1, 6 | wceq 1475 | . . 3 wff 𝐴 = ∪ 𝐴 |
8 | 3, 5, 7 | w3a 1031 | . 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) |
9 | 2, 8 | wb 195 | 1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: limeq 5652 dflim2 5698 limord 5701 limuni 5702 unizlim 5761 limon 6928 dflim3 6939 nnsuc 6974 onfununi 7325 dfrdg2 30945 ellimits 31187 onsucuni3 32391 |
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