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Definition df-pnf 7121
 Description: Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in ℝ and different from -∞ (df-mnf 7122). 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 7126 and mnfnre 7127, and we'll also be able to prove +∞ ≠ -∞. 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

Detailed syntax breakdown of Definition df-pnf
StepHypRef Expression
1 cpnf 7116 . 2 class +∞
2 cc 6945 . . . 4 class
32cuni 3608 . . 3 class
43cpw 3387 . 2 class 𝒫
51, 4wceq 1259 1 wff +∞ = 𝒫
 Colors of variables: wff set class This definition is referenced by:  pnfnre  7126  mnfnre  7127  pnfxr  8793
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