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Theorem ustn0 20591
Description: The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
ustn0  |-  -.  (/)  e.  U. ran UnifOn

Proof of Theorem ustn0
Dummy variables  v  u  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3794 . . . . 5  |-  -.  (
x  X.  x )  e.  (/)
2 0ex 4583 . . . . . 6  |-  (/)  e.  _V
3 eleq2 2540 . . . . . 6  |-  ( u  =  (/)  ->  ( ( x  X.  x )  e.  u  <->  ( x  X.  x )  e.  (/) ) )
42, 3elab 3255 . . . . 5  |-  ( (/)  e.  { u  |  ( x  X.  x )  e.  u }  <->  ( x  X.  x )  e.  (/) )
51, 4mtbir 299 . . . 4  |-  -.  (/)  e.  {
u  |  ( x  X.  x )  e.  u }
6 vex 3121 . . . . . . 7  |-  x  e. 
_V
7 selpw 4023 . . . . . . . . . 10  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
87abbii 2601 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
9 abid2 2607 . . . . . . . . . 10  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
106, 6xpex 6599 . . . . . . . . . . . 12  |-  ( x  X.  x )  e. 
_V
1110pwex 4636 . . . . . . . . . . 11  |-  ~P (
x  X.  x )  e.  _V
1211pwex 4636 . . . . . . . . . 10  |-  ~P ~P ( x  X.  x
)  e.  _V
139, 12eqeltri 2551 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
148, 13eqeltrri 2552 . . . . . . . 8  |-  { u  |  u  C_  ~P (
x  X.  x ) }  e.  _V
15 simp1 996 . . . . . . . . 9  |-  ( ( u  C_  ~P (
x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) )  ->  u  C_ 
~P ( x  X.  x ) )
1615ss2abi 3577 . . . . . . . 8  |-  { u  |  ( u  C_  ~P ( x  X.  x
)  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  C_  { u  |  u  C_  ~P ( x  X.  x
) }
1714, 16ssexi 4598 . . . . . . 7  |-  { u  |  ( u  C_  ~P ( x  X.  x
)  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  e.  _V
18 df-ust 20571 . . . . . . . 8  |- UnifOn  =  ( x  e.  _V  |->  { u  |  ( u 
C_  ~P ( x  X.  x )  /\  (
x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) } )
1918fvmpt2 5964 . . . . . . 7  |-  ( ( x  e.  _V  /\  { u  |  ( u 
C_  ~P ( x  X.  x )  /\  (
x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  e.  _V )  ->  (UnifOn `  x )  =  {
u  |  ( u 
C_  ~P ( x  X.  x )  /\  (
x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) } )
206, 17, 19mp2an 672 . . . . . 6  |-  (UnifOn `  x )  =  {
u  |  ( u 
C_  ~P ( x  X.  x )  /\  (
x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }
21 simp2 997 . . . . . . 7  |-  ( ( u  C_  ~P (
x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) )  ->  (
x  X.  x )  e.  u )
2221ss2abi 3577 . . . . . 6  |-  { u  |  ( u  C_  ~P ( x  X.  x
)  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  C_  { u  |  ( x  X.  x )  e.  u }
2320, 22eqsstri 3539 . . . . 5  |-  (UnifOn `  x )  C_  { u  |  ( x  X.  x )  e.  u }
2423sseli 3505 . . . 4  |-  ( (/)  e.  (UnifOn `  x )  -> 
(/)  e.  { u  |  ( x  X.  x )  e.  u } )
255, 24mto 176 . . 3  |-  -.  (/)  e.  (UnifOn `  x )
2625nex 1610 . 2  |-  -.  E. x (/)  e.  (UnifOn `  x )
2718funmpt2 5631 . . . 4  |-  Fun UnifOn
28 elunirn 6162 . . . 4  |-  ( Fun UnifOn  ->  ( (/)  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x ) ) )
2927, 28ax-mp 5 . . 3  |-  ( (/)  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x )
)
30 ustfn 20572 . . . . 5  |- UnifOn  Fn  _V
31 fndm 5686 . . . . 5  |-  (UnifOn  Fn  _V  ->  dom UnifOn  =  _V )
3230, 31ax-mp 5 . . . 4  |-  dom UnifOn  =  _V
3332rexeqi 3068 . . 3  |-  ( E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x )  <->  E. x  e.  _V  (/)  e.  (UnifOn `  x ) )
34 rexv 3133 . . 3  |-  ( E. x  e.  _V  (/)  e.  (UnifOn `  x )  <->  E. x (/) 
e.  (UnifOn `  x )
)
3529, 33, 343bitri 271 . 2  |-  ( (/)  e.  U. ran UnifOn  <->  E. x (/)  e.  (UnifOn `  x ) )
3626, 35mtbir 299 1  |-  -.  (/)  e.  U. ran UnifOn
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251    _I cid 4796    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006    |` cres 5007    o. ccom 5009   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  UnifOncust 20570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ust 20571
This theorem is referenced by: (None)
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