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Theorem ustn0 19937
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 3752 . . . . 5  |-  -.  (
x  X.  x )  e.  (/)
2 0ex 4533 . . . . . 6  |-  (/)  e.  _V
3 eleq2 2527 . . . . . 6  |-  ( u  =  (/)  ->  ( ( x  X.  x )  e.  u  <->  ( x  X.  x )  e.  (/) ) )
42, 3elab 3213 . . . . 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 3081 . . . . . . 7  |-  x  e. 
_V
7 selpw 3978 . . . . . . . . . 10  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
87abbii 2588 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
9 abid2 2594 . . . . . . . . . 10  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
106, 6xpex 6621 . . . . . . . . . . . 12  |-  ( x  X.  x )  e. 
_V
1110pwex 4586 . . . . . . . . . . 11  |-  ~P (
x  X.  x )  e.  _V
1211pwex 4586 . . . . . . . . . 10  |-  ~P ~P ( x  X.  x
)  e.  _V
139, 12eqeltri 2538 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
148, 13eqeltrri 2539 . . . . . . . 8  |-  { u  |  u  C_  ~P (
x  X.  x ) }  e.  _V
15 simp1 988 . . . . . . . . 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 3535 . . . . . . . 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 4548 . . . . . . 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 19917 . . . . . . . 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 5893 . . . . . . 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 989 . . . . . . 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 3535 . . . . . 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 3497 . . . . 5  |-  (UnifOn `  x )  C_  { u  |  ( x  X.  x )  e.  u }
2423sseli 3463 . . . 4  |-  ( (/)  e.  (UnifOn `  x )  -> 
(/)  e.  { u  |  ( x  X.  x )  e.  u } )
255, 24mto 176 . . 3  |-  -.  (/)  e.  (UnifOn `  x )
2625nex 1601 . 2  |-  -.  E. x (/)  e.  (UnifOn `  x )
2718funmpt2 5566 . . . 4  |-  Fun UnifOn
28 elunirn 6080 . . . 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 19918 . . . . 5  |- UnifOn  Fn  _V
31 fndm 5621 . . . . 5  |-  (UnifOn  Fn  _V  ->  dom UnifOn  =  _V )
3230, 31ax-mp 5 . . . 4  |-  dom UnifOn  =  _V
3332rexeqi 3028 . . 3  |-  ( E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x )  <->  E. x  e.  _V  (/)  e.  (UnifOn `  x ) )
34 rexv 3093 . . 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 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   _Vcvv 3078    i^i cin 3438    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   U.cuni 4202    _I cid 4742    X. cxp 4949   `'ccnv 4950   dom cdm 4951   ran crn 4952    |` cres 4953    o. ccom 4955   Fun wfun 5523    Fn wfn 5524   ` cfv 5529  UnifOncust 19916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-ust 19917
This theorem is referenced by: (None)
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