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Theorem ustn0 21228
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 3734 . . . . 5  |-  -.  (
x  X.  x )  e.  (/)
2 0ex 4534 . . . . . 6  |-  (/)  e.  _V
3 eleq2 2517 . . . . . 6  |-  ( u  =  (/)  ->  ( ( x  X.  x )  e.  u  <->  ( x  X.  x )  e.  (/) ) )
42, 3elab 3184 . . . . 5  |-  ( (/)  e.  { u  |  ( x  X.  x )  e.  u }  <->  ( x  X.  x )  e.  (/) )
51, 4mtbir 301 . . . 4  |-  -.  (/)  e.  {
u  |  ( x  X.  x )  e.  u }
6 vex 3047 . . . . . . 7  |-  x  e. 
_V
7 selpw 3957 . . . . . . . . . 10  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
87abbii 2566 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
9 abid2 2572 . . . . . . . . . 10  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
106, 6xpex 6592 . . . . . . . . . . . 12  |-  ( x  X.  x )  e. 
_V
1110pwex 4585 . . . . . . . . . . 11  |-  ~P (
x  X.  x )  e.  _V
1211pwex 4585 . . . . . . . . . 10  |-  ~P ~P ( x  X.  x
)  e.  _V
139, 12eqeltri 2524 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
148, 13eqeltrri 2525 . . . . . . . 8  |-  { u  |  u  C_  ~P (
x  X.  x ) }  e.  _V
15 simp1 1007 . . . . . . . . 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 3500 . . . . . . . 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 4547 . . . . . . 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 21208 . . . . . . . 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 5955 . . . . . . 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 677 . . . . . 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 1008 . . . . . . 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 3500 . . . . . 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 3461 . . . . 5  |-  (UnifOn `  x )  C_  { u  |  ( x  X.  x )  e.  u }
2423sseli 3427 . . . 4  |-  ( (/)  e.  (UnifOn `  x )  -> 
(/)  e.  { u  |  ( x  X.  x )  e.  u } )
255, 24mto 180 . . 3  |-  -.  (/)  e.  (UnifOn `  x )
2625nex 1677 . 2  |-  -.  E. x (/)  e.  (UnifOn `  x )
2718funmpt2 5618 . . . 4  |-  Fun UnifOn
28 elunirn 6154 . . . 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 21209 . . . . 5  |- UnifOn  Fn  _V
31 fndm 5673 . . . . 5  |-  (UnifOn  Fn  _V  ->  dom UnifOn  =  _V )
3230, 31ax-mp 5 . . . 4  |-  dom UnifOn  =  _V
3332rexeqi 2991 . . 3  |-  ( E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x )  <->  E. x  e.  _V  (/)  e.  (UnifOn `  x ) )
34 rexv 3061 . . 3  |-  ( E. x  e.  _V  (/)  e.  (UnifOn `  x )  <->  E. x (/) 
e.  (UnifOn `  x )
)
3529, 33, 343bitri 275 . 2  |-  ( (/)  e.  U. ran UnifOn  <->  E. x (/)  e.  (UnifOn `  x ) )
3626, 35mtbir 301 1  |-  -.  (/)  e.  U. ran UnifOn
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   {cab 2436   A.wral 2736   E.wrex 2737   _Vcvv 3044    i^i cin 3402    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197    _I cid 4743    X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834    |` cres 4835    o. ccom 4837   Fun wfun 5575    Fn wfn 5576   ` cfv 5581  UnifOncust 21207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-fv 5589  df-ust 21208
This theorem is referenced by: (None)
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