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Theorem ust0 20485
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 4577 . . . . . . . 8  |-  (/)  e.  _V
2 isust 20469 . . . . . . . 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 1011 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  ~P ( (/)  X.  (/) ) )
5 0xp 5080 . . . . . . . 8  |-  ( (/)  X.  (/) )  =  (/)
65pweqi 4014 . . . . . . 7  |-  ~P ( (/) 
X.  (/) )  =  ~P (/)
7 pw0 4174 . . . . . . 7  |-  ~P (/)  =  { (/)
}
86, 7eqtri 2496 . . . . . 6  |-  ~P ( (/) 
X.  (/) )  =  { (/)
}
94, 8syl6sseq 3550 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  { (/) } )
10 ustbasel 20472 . . . . . . 7  |-  ( u  e.  (UnifOn `  (/) )  -> 
( (/)  X.  (/) )  e.  u )
115, 10syl5eqelr 2560 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  (/) 
e.  u )
1211snssd 4172 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  { (/) }  C_  u
)
139, 12eqssd 3521 . . . 4  |-  ( u  e.  (UnifOn `  (/) )  ->  u  =  { (/) } )
14 elsn 4041 . . . 4  |-  ( u  e.  { { (/) } }  <->  u  =  { (/)
} )
1513, 14sylibr 212 . . 3  |-  ( u  e.  (UnifOn `  (/) )  ->  u  e.  { { (/) } } )
1615ssriv 3508 . 2  |-  (UnifOn `  (/) )  C_  { { (/) } }
178eqimss2i 3559 . . . 4  |-  { (/) } 
C_  ~P ( (/)  X.  (/) )
181snid 4055 . . . . 5  |-  (/)  e.  { (/)
}
195, 18eqeltri 2551 . . . 4  |-  ( (/)  X.  (/) )  e.  { (/) }
2018a1i 11 . . . . . 6  |-  ( (/)  C_  (/)  ->  (/)  e.  { (/) } )
218raleqi 3062 . . . . . . 7  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e.  {
(/) } ) )
22 sseq2 3526 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( (/)  C_  w  <->  (/)  C_  (/) ) )
23 eleq1 2539 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
2422, 23imbi12d 320 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (
(/)  C_  w  ->  w  e.  { (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) ) )
251, 24ralsn 4066 . . . . . . 7  |-  ( A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e. 
{ (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) )
2621, 25bitri 249 . . . . . 6  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) )
2720, 26mpbir 209 . . . . 5  |-  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } )
28 inidm 3707 . . . . . . 7  |-  ( (/)  i^i  (/) )  =  (/)
2928, 18eqeltri 2551 . . . . . 6  |-  ( (/)  i^i  (/) )  e.  { (/) }
30 ineq2 3694 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (/)  i^i  w )  =  (
(/)  i^i  (/) ) )
3130eleq1d 2536 . . . . . . 7  |-  ( w  =  (/)  ->  ( (
(/)  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } ) )
321, 31ralsn 4066 . . . . . 6  |-  ( A. w  e.  { (/) }  ( (/) 
i^i  w )  e. 
{ (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } )
3329, 32mpbir 209 . . . . 5  |-  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
}
34 res0 5278 . . . . . . 7  |-  (  _I  |`  (/) )  =  (/)
3534eqimssi 3558 . . . . . 6  |-  (  _I  |`  (/) )  C_  (/)
36 cnv0 5409 . . . . . . 7  |-  `' (/)  =  (/)
3736, 18eqeltri 2551 . . . . . 6  |-  `' (/)  e.  { (/) }
38 co02 5521 . . . . . . . 8  |-  ( (/)  o.  (/) )  =  (/)
3938eqimssi 3558 . . . . . . 7  |-  ( (/)  o.  (/) )  C_  (/)
40 id 22 . . . . . . . . . 10  |-  ( w  =  (/)  ->  w  =  (/) )
4140, 40coeq12d 5167 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  o.  w )  =  ( (/)  o.  (/) ) )
4241sseq1d 3531 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w  o.  w ) 
C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) ) )
431, 42rexsn 4067 . . . . . . 7  |-  ( E. w  e.  { (/) }  ( w  o.  w
)  C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) )
4439, 43mpbir 209 . . . . . 6  |-  E. w  e.  { (/) }  ( w  o.  w )  C_  (/)
4535, 37, 443pm3.2i 1174 . . . . 5  |-  ( (  _I  |`  (/) )  C_  (/) 
/\  `' (/)  e.  { (/)
}  /\  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) )
46 sseq1 3525 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v 
C_  w  <->  (/)  C_  w
) )
4746imbi1d 317 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  C_  w  ->  w  e.  { (/) } )  <-> 
( (/)  C_  w  ->  w  e.  { (/) } ) ) )
4847ralbidv 2903 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  { (/) } )  <->  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } ) ) )
49 ineq1 3693 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v  i^i  w )  =  ( (/)  i^i  w
) )
5049eleq1d 2536 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  w
)  e.  { (/) } ) )
5150ralbidv 2903 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  { (/) }  (
v  i^i  w )  e.  { (/) }  <->  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
} ) )
52 sseq2 3526 . . . . . . . 8  |-  ( v  =  (/)  ->  ( (  _I  |`  (/) )  C_  v 
<->  (  _I  |`  (/) )  C_  (/) ) )
53 cnveq 5176 . . . . . . . . 9  |-  ( v  =  (/)  ->  `' v  =  `' (/) )
5453eleq1d 2536 . . . . . . . 8  |-  ( v  =  (/)  ->  ( `' v  e.  { (/) }  <->  `' (/)  e.  { (/) } ) )
55 sseq2 3526 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ( w  o.  w ) 
C_  v  <->  ( w  o.  w )  C_  (/) ) )
5655rexbidv 2973 . . . . . . . 8  |-  ( v  =  (/)  ->  ( E. w  e.  { (/) }  ( w  o.  w
)  C_  v  <->  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) ) )
5752, 54, 563anbi123d 1299 . . . . . . 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 1299 . . . . . 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 4066 . . . . 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 1178 . . . 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 20469 . . . . 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 1178 . . 3  |-  { (/) }  e.  (UnifOn `  (/) )
64 snssi 4171 . . 3  |-  ( {
(/) }  e.  (UnifOn `  (/) )  ->  { { (/)
} }  C_  (UnifOn `  (/) ) )
6563, 64ax-mp 5 . 2  |-  { { (/)
} }  C_  (UnifOn `  (/) )
6616, 65eqssi 3520 1  |-  (UnifOn `  (/) )  =  { { (/)
} }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027    _I cid 4790    X. cxp 4997   `'ccnv 4998    |` cres 5001    o. ccom 5003   ` cfv 5588  UnifOncust 20465
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ust 20466
This theorem is referenced by:  isusp  20527
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