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Definition df-gru 8413
Description: A Grothendieck's universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, cross products etc. Grothendieck and alii, Séminaire de Géométrie Algébrique 4, Exposé I, p. 185. It was designed to give a precise meaning to the concepts of categories of sets, groups... (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
df-gru  |-  Univ  =  { u  |  ( Tr  u  /\  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  {
x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u )
) }
Distinct variable group:    x, u, y

Detailed syntax breakdown of Definition df-gru
StepHypRef Expression
1 cgru 8412 . 2  class  Univ
2 vu . . . . . 6  set  u
32cv 1622 . . . . 5  class  u
43wtr 4113 . . . 4  wff  Tr  u
5 vx . . . . . . . . 9  set  x
65cv 1622 . . . . . . . 8  class  x
76cpw 3625 . . . . . . 7  class  ~P x
87, 3wcel 1684 . . . . . 6  wff  ~P x  e.  u
9 vy . . . . . . . . . 10  set  y
109cv 1622 . . . . . . . . 9  class  y
116, 10cpr 3641 . . . . . . . 8  class  { x ,  y }
1211, 3wcel 1684 . . . . . . 7  wff  { x ,  y }  e.  u
1312, 9, 3wral 2543 . . . . . 6  wff  A. y  e.  u  { x ,  y }  e.  u
1410crn 4690 . . . . . . . . 9  class  ran  y
1514cuni 3827 . . . . . . . 8  class  U. ran  y
1615, 3wcel 1684 . . . . . . 7  wff  U. ran  y  e.  u
17 cmap 6772 . . . . . . . 8  class  ^m
183, 6, 17co 5858 . . . . . . 7  class  ( u  ^m  x )
1916, 9, 18wral 2543 . . . . . 6  wff  A. y  e.  ( u  ^m  x
) U. ran  y  e.  u
208, 13, 19w3a 934 . . . . 5  wff  ( ~P x  e.  u  /\  A. y  e.  u  {
x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u )
2120, 5, 3wral 2543 . . . 4  wff  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  {
x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u )
224, 21wa 358 . . 3  wff  ( Tr  u  /\  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  {
x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u )
)
2322, 2cab 2269 . 2  class  { u  |  ( Tr  u  /\  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u
) ) }
241, 23wceq 1623 1  wff  Univ  =  { u  |  ( Tr  u  /\  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  {
x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u )
) }
Colors of variables: wff set class
This definition is referenced by:  elgrug  8414
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