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Definition df-goeq 23941
Description: Define the Godel-set of equality. Here the arguments  x  =  <. N ,  P >. correspond to vN and vP , so  ( (/)  =g  1o ) actually means v0  = v1 , not  0  = 
1. Here we use the trick mentioned in ax-ext 2264 to introduce equality as a defined notion in terms of  e.g. The expression  suc  ( u  u.  v )  = max  ( u ,  v )  +  1 here is a convenient way of getting a dummy variable distinct from  u and  v. (Contributed by Mario Carneiro, 14-Jul-2013.)
Assertion
Ref Expression
df-goeq  |-  =g  =  ( u  e.  om ,  v  e.  om  |->  [_
suc  ( u  u.  v )  /  w ]_ A.g w ( ( w  e.g  u ) 
<->g  ( w  e.g  v
) ) )
Distinct variable group:    v, u, w

Detailed syntax breakdown of Definition df-goeq
StepHypRef Expression
1 cgoq 23934 . 2  class  =g
2 vu . . 3  set  u
3 vv . . 3  set  v
4 com 4656 . . 3  class  om
5 vw . . . 4  set  w
62cv 1622 . . . . . 6  class  u
73cv 1622 . . . . . 6  class  v
86, 7cun 3150 . . . . 5  class  ( u  u.  v )
98csuc 4394 . . . 4  class  suc  (
u  u.  v )
105cv 1622 . . . . . . 7  class  w
11 cgoe 23916 . . . . . . 7  class  e.g
1210, 6, 11co 5858 . . . . . 6  class  ( w  e.g  u )
1310, 7, 11co 5858 . . . . . 6  class  ( w  e.g  v )
14 cgob 23933 . . . . . 6  class  <->g
1512, 13, 14co 5858 . . . . 5  class  ( ( w  e.g  u ) 
<->g  ( w  e.g  v
) )
1615, 10cgol 23918 . . . 4  class  A.g w
( ( w  e.g  u )  <->g  ( w  e.g  v ) )
175, 9, 16csb 3081 . . 3  class  [_ suc  ( u  u.  v
)  /  w ]_ A.g w ( ( w  e.g  u )  <->g  (
w  e.g  v )
)
182, 3, 4, 4, 17cmpt2 5860 . 2  class  ( u  e.  om ,  v  e.  om  |->  [_ suc  ( u  u.  v
)  /  w ]_ A.g w ( ( w  e.g  u )  <->g  (
w  e.g  v )
) )
191, 18wceq 1623 1  wff  =g  =  ( u  e.  om ,  v  e.  om  |->  [_
suc  ( u  u.  v )  /  w ]_ A.g w ( ( w  e.g  u ) 
<->g  ( w  e.g  v
) ) )
Colors of variables: wff set class
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