Definition df-lim 6390

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 6442, dflim3 7867, 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 6386	. 2 wff Lim 𝐴
3	1	word 6384	. . 3 wff Ord 𝐴
4		c0 4338	. . . 4 class ∅
5	1, 4	wne 2937	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4911	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1536	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1086	. 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)
9	2, 8	wb 206	1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

This definition is referenced by:  limeq  6397  dflim2  6442  limord  6445  limuni  6446  unizlim  6508  limon  7855  dflim3  7867  nnsuc  7904  onfununi  8379  nlim1  8525  nlim2  8526  dfrdg2  35776  ellimits  35891  onsucuni3  37349  omlimcl2  43230  dflim5  43318