Definition df-lim 6389

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 6441, dflim3 7868, 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 6385	. 2 wff Lim 𝐴
3	1	word 6383	. . 3 wff Ord 𝐴
4		c0 4333	. . . 4 class ∅
5	1, 4	wne 2940	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4907	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1540	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1087	. 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  6396  dflim2  6441  limord  6444  limuni  6445  unizlim  6507  limon  7856  dflim3  7868  nnsuc  7905  onfununi  8381  nlim1  8527  nlim2  8528  dfrdg2  35796  ellimits  35911  onsucuni3  37368  omlimcl2  43254  dflim5  43342