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Theorem ustn0 20849
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 3797 . . . . 5  |-  -.  (
x  X.  x )  e.  (/)
2 0ex 4587 . . . . . 6  |-  (/)  e.  _V
3 eleq2 2530 . . . . . 6  |-  ( u  =  (/)  ->  ( ( x  X.  x )  e.  u  <->  ( x  X.  x )  e.  (/) ) )
42, 3elab 3246 . . . . 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 3112 . . . . . . 7  |-  x  e. 
_V
7 selpw 4022 . . . . . . . . . 10  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
87abbii 2591 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
9 abid2 2597 . . . . . . . . . 10  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
106, 6xpex 6603 . . . . . . . . . . . 12  |-  ( x  X.  x )  e. 
_V
1110pwex 4639 . . . . . . . . . . 11  |-  ~P (
x  X.  x )  e.  _V
1211pwex 4639 . . . . . . . . . 10  |-  ~P ~P ( x  X.  x
)  e.  _V
139, 12eqeltri 2541 . . . . . . . . 9  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
148, 13eqeltrri 2542 . . . . . . . 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 3568 . . . . . . . 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 4601 . . . . . . 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 20829 . . . . . . . 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 3568 . . . . . 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 3529 . . . . 5  |-  (UnifOn `  x )  C_  { u  |  ( x  X.  x )  e.  u }
2423sseli 3495 . . . 4  |-  ( (/)  e.  (UnifOn `  x )  -> 
(/)  e.  { u  |  ( x  X.  x )  e.  u } )
255, 24mto 176 . . 3  |-  -.  (/)  e.  (UnifOn `  x )
2625nex 1628 . 2  |-  -.  E. x (/)  e.  (UnifOn `  x )
2718funmpt2 5631 . . . 4  |-  Fun UnifOn
28 elunirn 6164 . . . 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 20830 . . . . 5  |- UnifOn  Fn  _V
31 fndm 5686 . . . . 5  |-  (UnifOn  Fn  _V  ->  dom UnifOn  =  _V )
3230, 31ax-mp 5 . . . 4  |-  dom UnifOn  =  _V
3332rexeqi 3059 . . 3  |-  ( E. x  e.  dom UnifOn (/)  e.  (UnifOn `  x )  <->  E. x  e.  _V  (/)  e.  (UnifOn `  x ) )
34 rexv 3124 . . 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 1395   E.wex 1613    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251    _I cid 4799    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010    o. ccom 5012   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  UnifOncust 20828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ust 20829
This theorem is referenced by: (None)
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