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Theorem ust0 21227
Description: The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ust0  |-  (UnifOn `  (/) )  =  { { (/)
} }

Proof of Theorem ust0
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4534 . . . . . . . 8  |-  (/)  e.  _V
2 isust 21211 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( u  e.  (UnifOn `  (/) )  <->  ( u  C_ 
~P ( (/)  X.  (/) )  /\  ( (/)  X.  (/) )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  u )  /\  A. w  e.  u  (
v  i^i  w )  e.  u  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w )  C_  v
) ) ) ) )
31, 2ax-mp 5 . . . . . . 7  |-  ( u  e.  (UnifOn `  (/) )  <->  ( u  C_ 
~P ( (/)  X.  (/) )  /\  ( (/)  X.  (/) )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  u )  /\  A. w  e.  u  (
v  i^i  w )  e.  u  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w )  C_  v
) ) ) )
43simp1bi 1022 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  ~P ( (/)  X.  (/) ) )
5 0xp 4914 . . . . . . . 8  |-  ( (/)  X.  (/) )  =  (/)
65pweqi 3954 . . . . . . 7  |-  ~P ( (/) 
X.  (/) )  =  ~P (/)
7 pw0 4118 . . . . . . 7  |-  ~P (/)  =  { (/)
}
86, 7eqtri 2472 . . . . . 6  |-  ~P ( (/) 
X.  (/) )  =  { (/)
}
94, 8syl6sseq 3477 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  { (/) } )
10 ustbasel 21214 . . . . . . 7  |-  ( u  e.  (UnifOn `  (/) )  -> 
( (/)  X.  (/) )  e.  u )
115, 10syl5eqelr 2533 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  (/) 
e.  u )
1211snssd 4116 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  { (/) }  C_  u
)
139, 12eqssd 3448 . . . 4  |-  ( u  e.  (UnifOn `  (/) )  ->  u  =  { (/) } )
14 elsn 3981 . . . 4  |-  ( u  e.  { { (/) } }  <->  u  =  { (/)
} )
1513, 14sylibr 216 . . 3  |-  ( u  e.  (UnifOn `  (/) )  ->  u  e.  { { (/) } } )
1615ssriv 3435 . 2  |-  (UnifOn `  (/) )  C_  { { (/) } }
178eqimss2i 3486 . . . 4  |-  { (/) } 
C_  ~P ( (/)  X.  (/) )
181snid 3995 . . . . 5  |-  (/)  e.  { (/)
}
195, 18eqeltri 2524 . . . 4  |-  ( (/)  X.  (/) )  e.  { (/) }
2018a1i 11 . . . . . 6  |-  ( (/)  C_  (/)  ->  (/)  e.  { (/) } )
218raleqi 2990 . . . . . . 7  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e.  {
(/) } ) )
22 sseq2 3453 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( (/)  C_  w  <->  (/)  C_  (/) ) )
23 eleq1 2516 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
2422, 23imbi12d 322 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (
(/)  C_  w  ->  w  e.  { (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) ) )
251, 24ralsn 4009 . . . . . . 7  |-  ( A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e. 
{ (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) )
2621, 25bitri 253 . . . . . 6  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) )
2720, 26mpbir 213 . . . . 5  |-  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } )
28 inidm 3640 . . . . . . 7  |-  ( (/)  i^i  (/) )  =  (/)
2928, 18eqeltri 2524 . . . . . 6  |-  ( (/)  i^i  (/) )  e.  { (/) }
30 ineq2 3627 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (/)  i^i  w )  =  (
(/)  i^i  (/) ) )
3130eleq1d 2512 . . . . . . 7  |-  ( w  =  (/)  ->  ( (
(/)  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } ) )
321, 31ralsn 4009 . . . . . 6  |-  ( A. w  e.  { (/) }  ( (/) 
i^i  w )  e. 
{ (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } )
3329, 32mpbir 213 . . . . 5  |-  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
}
34 res0 5108 . . . . . . 7  |-  (  _I  |`  (/) )  =  (/)
3534eqimssi 3485 . . . . . 6  |-  (  _I  |`  (/) )  C_  (/)
36 cnv0 5238 . . . . . . 7  |-  `' (/)  =  (/)
3736, 18eqeltri 2524 . . . . . 6  |-  `' (/)  e.  { (/) }
38 co02 5348 . . . . . . . 8  |-  ( (/)  o.  (/) )  =  (/)
3938eqimssi 3485 . . . . . . 7  |-  ( (/)  o.  (/) )  C_  (/)
40 id 22 . . . . . . . . . 10  |-  ( w  =  (/)  ->  w  =  (/) )
4140, 40coeq12d 4998 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  o.  w )  =  ( (/)  o.  (/) ) )
4241sseq1d 3458 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w  o.  w ) 
C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) ) )
431, 42rexsn 4010 . . . . . . 7  |-  ( E. w  e.  { (/) }  ( w  o.  w
)  C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) )
4439, 43mpbir 213 . . . . . 6  |-  E. w  e.  { (/) }  ( w  o.  w )  C_  (/)
4535, 37, 443pm3.2i 1185 . . . . 5  |-  ( (  _I  |`  (/) )  C_  (/) 
/\  `' (/)  e.  { (/)
}  /\  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) )
46 sseq1 3452 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v 
C_  w  <->  (/)  C_  w
) )
4746imbi1d 319 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  C_  w  ->  w  e.  { (/) } )  <-> 
( (/)  C_  w  ->  w  e.  { (/) } ) ) )
4847ralbidv 2826 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  { (/) } )  <->  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } ) ) )
49 ineq1 3626 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v  i^i  w )  =  ( (/)  i^i  w
) )
5049eleq1d 2512 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  w
)  e.  { (/) } ) )
5150ralbidv 2826 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  { (/) }  (
v  i^i  w )  e.  { (/) }  <->  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
} ) )
52 sseq2 3453 . . . . . . . 8  |-  ( v  =  (/)  ->  ( (  _I  |`  (/) )  C_  v 
<->  (  _I  |`  (/) )  C_  (/) ) )
53 cnveq 5007 . . . . . . . . 9  |-  ( v  =  (/)  ->  `' v  =  `' (/) )
5453eleq1d 2512 . . . . . . . 8  |-  ( v  =  (/)  ->  ( `' v  e.  { (/) }  <->  `' (/)  e.  { (/) } ) )
55 sseq2 3453 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ( w  o.  w ) 
C_  v  <->  ( w  o.  w )  C_  (/) ) )
5655rexbidv 2900 . . . . . . . 8  |-  ( v  =  (/)  ->  ( E. w  e.  { (/) }  ( w  o.  w
)  C_  v  <->  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) ) )
5752, 54, 563anbi123d 1338 . . . . . . 7  |-  ( v  =  (/)  ->  ( ( (  _I  |`  (/) )  C_  v  /\  `' v  e. 
{ (/) }  /\  E. w  e.  { (/) }  (
w  o.  w ) 
C_  v )  <->  ( (  _I  |`  (/) )  C_  (/)  /\  `' (/) 
e.  { (/) }  /\  E. w  e.  { (/) }  ( w  o.  w
)  C_  (/) ) ) )
5848, 51, 573anbi123d 1338 . . . . . 6  |-  ( v  =  (/)  ->  ( ( A. w  e.  ~P  ( (/)  X.  (/) ) ( v  C_  w  ->  w  e.  { (/) } )  /\  A. w  e. 
{ (/) }  ( v  i^i  w )  e. 
{ (/) }  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e. 
{ (/) }  /\  E. w  e.  { (/) }  (
w  o.  w ) 
C_  v ) )  <-> 
( A. w  e. 
~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } )  /\  A. w  e.  { (/) }  ( (/)  i^i  w
)  e.  { (/) }  /\  ( (  _I  |`  (/) )  C_  (/)  /\  `' (/) 
e.  { (/) }  /\  E. w  e.  { (/) }  ( w  o.  w
)  C_  (/) ) ) ) )
591, 58ralsn 4009 . . . . 5  |-  ( A. v  e.  { (/) }  ( A. w  e.  ~P  ( (/)  X.  (/) ) ( v  C_  w  ->  w  e.  { (/) } )  /\  A. w  e. 
{ (/) }  ( v  i^i  w )  e. 
{ (/) }  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e. 
{ (/) }  /\  E. w  e.  { (/) }  (
w  o.  w ) 
C_  v ) )  <-> 
( A. w  e. 
~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } )  /\  A. w  e.  { (/) }  ( (/)  i^i  w
)  e.  { (/) }  /\  ( (  _I  |`  (/) )  C_  (/)  /\  `' (/) 
e.  { (/) }  /\  E. w  e.  { (/) }  ( w  o.  w
)  C_  (/) ) ) )
6027, 33, 45, 59mpbir3an 1189 . . . 4  |-  A. v  e.  { (/) }  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  { (/) } )  /\  A. w  e.  { (/) }  ( v  i^i  w
)  e.  { (/) }  /\  ( (  _I  |`  (/) )  C_  v  /\  `' v  e.  { (/) }  /\  E. w  e. 
{ (/) }  ( w  o.  w )  C_  v ) )
61 isust 21211 . . . . 5  |-  ( (/)  e.  _V  ->  ( { (/)
}  e.  (UnifOn `  (/) )  <->  ( { (/) } 
C_  ~P ( (/)  X.  (/) )  /\  ( (/)  X.  (/) )  e. 
{ (/) }  /\  A. v  e.  { (/) }  ( A. w  e.  ~P  ( (/)  X.  (/) ) ( v  C_  w  ->  w  e.  { (/) } )  /\  A. w  e. 
{ (/) }  ( v  i^i  w )  e. 
{ (/) }  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e. 
{ (/) }  /\  E. w  e.  { (/) }  (
w  o.  w ) 
C_  v ) ) ) ) )
621, 61ax-mp 5 . . . 4  |-  ( {
(/) }  e.  (UnifOn `  (/) )  <->  ( { (/) } 
C_  ~P ( (/)  X.  (/) )  /\  ( (/)  X.  (/) )  e. 
{ (/) }  /\  A. v  e.  { (/) }  ( A. w  e.  ~P  ( (/)  X.  (/) ) ( v  C_  w  ->  w  e.  { (/) } )  /\  A. w  e. 
{ (/) }  ( v  i^i  w )  e. 
{ (/) }  /\  (
(  _I  |`  (/) )  C_  v  /\  `' v  e. 
{ (/) }  /\  E. w  e.  { (/) }  (
w  o.  w ) 
C_  v ) ) ) )
6317, 19, 60, 62mpbir3an 1189 . . 3  |-  { (/) }  e.  (UnifOn `  (/) )
64 snssi 4115 . . 3  |-  ( {
(/) }  e.  (UnifOn `  (/) )  ->  { { (/)
} }  C_  (UnifOn `  (/) ) )
6563, 64ax-mp 5 . 2  |-  { { (/)
} }  C_  (UnifOn `  (/) )
6616, 65eqssi 3447 1  |-  (UnifOn `  (/) )  =  { { (/)
} }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   _Vcvv 3044    i^i cin 3402    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   {csn 3967    _I cid 4743    X. cxp 4831   `'ccnv 4832    |` cres 4835    o. ccom 4837   ` 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-res 4845  df-iota 5545  df-fun 5583  df-fv 5589  df-ust 21208
This theorem is referenced by:  isusp  21269
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