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

Definition df-ina 8323
Description: An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
df-ina  |-  Inacc  =  {
x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) }
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-ina
StepHypRef Expression
1 cina 8321 . 2  class  Inacc
2 vx . . . . . 6  set  x
32cv 1631 . . . . 5  class  x
4 c0 3468 . . . . 5  class  (/)
53, 4wne 2459 . . . 4  wff  x  =/=  (/)
6 ccf 7586 . . . . . 6  class  cf
73, 6cfv 5271 . . . . 5  class  ( cf `  x )
87, 3wceq 1632 . . . 4  wff  ( cf `  x )  =  x
9 vy . . . . . . . 8  set  y
109cv 1631 . . . . . . 7  class  y
1110cpw 3638 . . . . . 6  class  ~P y
12 csdm 6878 . . . . . 6  class  ~<
1311, 3, 12wbr 4039 . . . . 5  wff  ~P y  ~<  x
1413, 9, 3wral 2556 . . . 4  wff  A. y  e.  x  ~P y  ~<  x
155, 8, 14w3a 934 . . 3  wff  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x )
1615, 2cab 2282 . 2  class  { x  |  ( x  =/=  (/)  /\  ( cf `  x
)  =  x  /\  A. y  e.  x  ~P y  ~<  x ) }
171, 16wceq 1632 1  wff  Inacc  =  {
x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) }
Colors of variables: wff set class
This definition is referenced by:  elina  8325
  Copyright terms: Public domain W3C validator