Definition df-pnf 10917

Description: Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in ℝ and different from -∞ (df-mnf 10918). We use 𝒫 ∪ ℂ to make it independent of the construction of ℂ, and Cantor's Theorem will show that it is different from any member of ℂ and therefore ℝ. See pnfnre 10922, mnfnre 10924, and pnfnemnf 10936.

A simpler possibility is to define +∞ as ℂ and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of ℝ. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
df-pnf	⊢ +∞ = 𝒫 ∪ ℂ

Detailed syntax breakdown of Definition df-pnf

Step	Hyp	Ref	Expression
1		cpnf 10912	. 2 class +∞
2		cc 10775	. . . 4 class ℂ
3	2	cuni 4836	. . 3 class ∪ ℂ
4	3	cpw 4530	. 2 class 𝒫 ∪ ℂ
5	1, 4	wceq 1543	1 wff +∞ = 𝒫 ∪ ℂ

This definition is referenced by:  pnfnre  10922  mnfnre  10924  pnfex  10934