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Theorem ust0 19792
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 4420 . . . . . . . 8  |-  (/)  e.  _V
2 isust 19776 . . . . . . . 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 1003 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  ~P ( (/)  X.  (/) ) )
5 0xp 4915 . . . . . . . 8  |-  ( (/)  X.  (/) )  =  (/)
65pweqi 3862 . . . . . . 7  |-  ~P ( (/) 
X.  (/) )  =  ~P (/)
7 pw0 4018 . . . . . . 7  |-  ~P (/)  =  { (/)
}
86, 7eqtri 2461 . . . . . 6  |-  ~P ( (/) 
X.  (/) )  =  { (/)
}
94, 8syl6sseq 3400 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  { (/) } )
10 ustbasel 19779 . . . . . . 7  |-  ( u  e.  (UnifOn `  (/) )  -> 
( (/)  X.  (/) )  e.  u )
115, 10syl5eqelr 2526 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  (/) 
e.  u )
1211snssd 4016 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  { (/) }  C_  u
)
139, 12eqssd 3371 . . . 4  |-  ( u  e.  (UnifOn `  (/) )  ->  u  =  { (/) } )
14 elsn 3889 . . . 4  |-  ( u  e.  { { (/) } }  <->  u  =  { (/)
} )
1513, 14sylibr 212 . . 3  |-  ( u  e.  (UnifOn `  (/) )  ->  u  e.  { { (/) } } )
1615ssriv 3358 . 2  |-  (UnifOn `  (/) )  C_  { { (/) } }
178eqimss2i 3409 . . . 4  |-  { (/) } 
C_  ~P ( (/)  X.  (/) )
181snid 3903 . . . . 5  |-  (/)  e.  { (/)
}
195, 18eqeltri 2511 . . . 4  |-  ( (/)  X.  (/) )  e.  { (/) }
2018a1i 11 . . . . . 6  |-  ( (/)  C_  (/)  ->  (/)  e.  { (/) } )
218raleqi 2919 . . . . . . 7  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e.  {
(/) } ) )
22 sseq2 3376 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( (/)  C_  w  <->  (/)  C_  (/) ) )
23 eleq1 2501 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
2422, 23imbi12d 320 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (
(/)  C_  w  ->  w  e.  { (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) ) )
251, 24ralsn 3913 . . . . . . 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 3557 . . . . . . 7  |-  ( (/)  i^i  (/) )  =  (/)
2928, 18eqeltri 2511 . . . . . 6  |-  ( (/)  i^i  (/) )  e.  { (/) }
30 ineq2 3544 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (/)  i^i  w )  =  (
(/)  i^i  (/) ) )
3130eleq1d 2507 . . . . . . 7  |-  ( w  =  (/)  ->  ( (
(/)  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } ) )
321, 31ralsn 3913 . . . . . 6  |-  ( A. w  e.  { (/) }  ( (/) 
i^i  w )  e. 
{ (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } )
3329, 32mpbir 209 . . . . 5  |-  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
}
34 res0 5113 . . . . . . 7  |-  (  _I  |`  (/) )  =  (/)
3534eqimssi 3408 . . . . . 6  |-  (  _I  |`  (/) )  C_  (/)
36 cnv0 5238 . . . . . . 7  |-  `' (/)  =  (/)
3736, 18eqeltri 2511 . . . . . 6  |-  `' (/)  e.  { (/) }
38 co02 5349 . . . . . . . 8  |-  ( (/)  o.  (/) )  =  (/)
3938eqimssi 3408 . . . . . . 7  |-  ( (/)  o.  (/) )  C_  (/)
40 id 22 . . . . . . . . . 10  |-  ( w  =  (/)  ->  w  =  (/) )
4140, 40coeq12d 5002 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  o.  w )  =  ( (/)  o.  (/) ) )
4241sseq1d 3381 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w  o.  w ) 
C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) ) )
431, 42rexsn 3914 . . . . . . 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 1166 . . . . 5  |-  ( (  _I  |`  (/) )  C_  (/) 
/\  `' (/)  e.  { (/)
}  /\  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) )
46 sseq1 3375 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v 
C_  w  <->  (/)  C_  w
) )
4746imbi1d 317 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  C_  w  ->  w  e.  { (/) } )  <-> 
( (/)  C_  w  ->  w  e.  { (/) } ) ) )
4847ralbidv 2733 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  { (/) } )  <->  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } ) ) )
49 ineq1 3543 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v  i^i  w )  =  ( (/)  i^i  w
) )
5049eleq1d 2507 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  w
)  e.  { (/) } ) )
5150ralbidv 2733 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  { (/) }  (
v  i^i  w )  e.  { (/) }  <->  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
} ) )
52 sseq2 3376 . . . . . . . 8  |-  ( v  =  (/)  ->  ( (  _I  |`  (/) )  C_  v 
<->  (  _I  |`  (/) )  C_  (/) ) )
53 cnveq 5011 . . . . . . . . 9  |-  ( v  =  (/)  ->  `' v  =  `' (/) )
5453eleq1d 2507 . . . . . . . 8  |-  ( v  =  (/)  ->  ( `' v  e.  { (/) }  <->  `' (/)  e.  { (/) } ) )
55 sseq2 3376 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ( w  o.  w ) 
C_  v  <->  ( w  o.  w )  C_  (/) ) )
5655rexbidv 2734 . . . . . . . 8  |-  ( v  =  (/)  ->  ( E. w  e.  { (/) }  ( w  o.  w
)  C_  v  <->  E. w  e.  { (/) }  ( w  o.  w )  C_  (/) ) )
5752, 54, 563anbi123d 1289 . . . . . . 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 1289 . . . . . 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 3913 . . . . 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 1170 . . . 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 19776 . . . . 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 1170 . . 3  |-  { (/) }  e.  (UnifOn `  (/) )
64 snssi 4015 . . 3  |-  ( {
(/) }  e.  (UnifOn `  (/) )  ->  { { (/)
} }  C_  (UnifOn `  (/) ) )
6563, 64ax-mp 5 . 2  |-  { { (/)
} }  C_  (UnifOn `  (/) )
6616, 65eqssi 3370 1  |-  (UnifOn `  (/) )  =  { { (/)
} }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3325    C_ wss 3326   (/)c0 3635   ~Pcpw 3858   {csn 3875    _I cid 4629    X. cxp 4836   `'ccnv 4837    |` cres 4840    o. ccom 4842   ` cfv 5416  UnifOncust 19772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-iota 5379  df-fun 5418  df-fv 5424  df-ust 19773
This theorem is referenced by:  isusp  19834
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