MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-pnf Unicode version

Definition df-pnf 9106
Description: Define plus infinity. Note that the definition is arbitrary, requiring only that  +oo be a set not in  RR and different from  -oo (df-mnf 9107). We use  ~P
U. CC to make it independent of the construction of  CC, and Cantor's Theorem will show that it is different from any member of 
CC and therefore  RR. See pnfnre 9111, mnfnre 9112, and pnfnemnf 10701.

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

Assertion
Ref Expression
df-pnf  |-  +oo  =  ~P U. CC

Detailed syntax breakdown of Definition df-pnf
StepHypRef Expression
1 cpnf 9101 . 2  class  +oo
2 cc 8972 . . . 4  class  CC
32cuni 4002 . . 3  class  U. CC
43cpw 3786 . 2  class  ~P U. CC
51, 4wceq 1652 1  wff  +oo  =  ~P U. CC
Colors of variables: wff set class
This definition is referenced by:  pnfnre  9111  mnfnre  9112  pnfxr  10697
  Copyright terms: Public domain W3C validator