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Definition df-wina 8306
Description: An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows  om as a weakly inacessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
df-wina  |-  Inacc W  =  { x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z ) }
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-wina
StepHypRef Expression
1 cwina 8304 . 2  class  Inacc W
2 vx . . . . . 6  set  x
32cv 1622 . . . . 5  class  x
4 c0 3455 . . . . 5  class  (/)
53, 4wne 2446 . . . 4  wff  x  =/=  (/)
6 ccf 7570 . . . . . 6  class  cf
73, 6cfv 5255 . . . . 5  class  ( cf `  x )
87, 3wceq 1623 . . . 4  wff  ( cf `  x )  =  x
9 vy . . . . . . . 8  set  y
109cv 1622 . . . . . . 7  class  y
11 vz . . . . . . . 8  set  z
1211cv 1622 . . . . . . 7  class  z
13 csdm 6862 . . . . . . 7  class  ~<
1410, 12, 13wbr 4023 . . . . . 6  wff  y  ~< 
z
1514, 11, 3wrex 2544 . . . . 5  wff  E. z  e.  x  y  ~<  z
1615, 9, 3wral 2543 . . . 4  wff  A. y  e.  x  E. z  e.  x  y  ~<  z
175, 8, 16w3a 934 . . 3  wff  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
)
1817, 2cab 2269 . 2  class  { x  |  ( x  =/=  (/)  /\  ( cf `  x
)  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z ) }
191, 18wceq 1623 1  wff  Inacc W  =  { x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z ) }
Colors of variables: wff set class
This definition is referenced by:  elwina  8308
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