Definition df-lim 5887

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 5940, dflim3 7210, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)

Assertion

Ref	Expression
df-lim	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

Detailed syntax breakdown of Definition df-lim

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	wlim 5883	. 2 wff Lim 𝐴
3	1	word 5881	. . 3 wff Ord 𝐴
4		c0 4056	. . . 4 class ∅
5	1, 4	wne 2930	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4586	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1630	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1072	. 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)
9	2, 8	wb 196	1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

This definition is referenced by:  limeq  5894  dflim2  5940  limord  5943  limuni  5944  unizlim  6003  limon  7199  dflim3  7210  nnsuc  7245  onfununi  7605  dfrdg2  32004  ellimits  32321  onsucuni3  33524