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Theorem ust0 20847
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 4587 . . . . . . . 8  |-  (/)  e.  _V
2 isust 20831 . . . . . . . 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 5089 . . . . . . . 8  |-  ( (/)  X.  (/) )  =  (/)
65pweqi 4019 . . . . . . 7  |-  ~P ( (/) 
X.  (/) )  =  ~P (/)
7 pw0 4179 . . . . . . 7  |-  ~P (/)  =  { (/)
}
86, 7eqtri 2486 . . . . . 6  |-  ~P ( (/) 
X.  (/) )  =  { (/)
}
94, 8syl6sseq 3545 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  u  C_  { (/) } )
10 ustbasel 20834 . . . . . . 7  |-  ( u  e.  (UnifOn `  (/) )  -> 
( (/)  X.  (/) )  e.  u )
115, 10syl5eqelr 2550 . . . . . 6  |-  ( u  e.  (UnifOn `  (/) )  ->  (/) 
e.  u )
1211snssd 4177 . . . . 5  |-  ( u  e.  (UnifOn `  (/) )  ->  { (/) }  C_  u
)
139, 12eqssd 3516 . . . 4  |-  ( u  e.  (UnifOn `  (/) )  ->  u  =  { (/) } )
14 elsn 4046 . . . 4  |-  ( u  e.  { { (/) } }  <->  u  =  { (/)
} )
1513, 14sylibr 212 . . 3  |-  ( u  e.  (UnifOn `  (/) )  ->  u  e.  { { (/) } } )
1615ssriv 3503 . 2  |-  (UnifOn `  (/) )  C_  { { (/) } }
178eqimss2i 3554 . . . 4  |-  { (/) } 
C_  ~P ( (/)  X.  (/) )
181snid 4060 . . . . 5  |-  (/)  e.  { (/)
}
195, 18eqeltri 2541 . . . 4  |-  ( (/)  X.  (/) )  e.  { (/) }
2018a1i 11 . . . . . 6  |-  ( (/)  C_  (/)  ->  (/)  e.  { (/) } )
218raleqi 3058 . . . . . . 7  |-  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( (/)  C_  w  ->  w  e.  {
(/) } )  <->  A. w  e.  { (/) }  ( (/)  C_  w  ->  w  e.  {
(/) } ) )
22 sseq2 3521 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( (/)  C_  w  <->  (/)  C_  (/) ) )
23 eleq1 2529 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
2422, 23imbi12d 320 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (
(/)  C_  w  ->  w  e.  { (/) } )  <->  ( (/)  C_  (/)  ->  (/)  e.  { (/)
} ) ) )
251, 24ralsn 4071 . . . . . . 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 3703 . . . . . . 7  |-  ( (/)  i^i  (/) )  =  (/)
2928, 18eqeltri 2541 . . . . . 6  |-  ( (/)  i^i  (/) )  e.  { (/) }
30 ineq2 3690 . . . . . . . 8  |-  ( w  =  (/)  ->  ( (/)  i^i  w )  =  (
(/)  i^i  (/) ) )
3130eleq1d 2526 . . . . . . 7  |-  ( w  =  (/)  ->  ( (
(/)  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } ) )
321, 31ralsn 4071 . . . . . 6  |-  ( A. w  e.  { (/) }  ( (/) 
i^i  w )  e. 
{ (/) }  <->  ( (/)  i^i  (/) )  e. 
{ (/) } )
3329, 32mpbir 209 . . . . 5  |-  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
}
34 res0 5288 . . . . . . 7  |-  (  _I  |`  (/) )  =  (/)
3534eqimssi 3553 . . . . . 6  |-  (  _I  |`  (/) )  C_  (/)
36 cnv0 5416 . . . . . . 7  |-  `' (/)  =  (/)
3736, 18eqeltri 2541 . . . . . 6  |-  `' (/)  e.  { (/) }
38 co02 5527 . . . . . . . 8  |-  ( (/)  o.  (/) )  =  (/)
3938eqimssi 3553 . . . . . . 7  |-  ( (/)  o.  (/) )  C_  (/)
40 id 22 . . . . . . . . . 10  |-  ( w  =  (/)  ->  w  =  (/) )
4140, 40coeq12d 5177 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w  o.  w )  =  ( (/)  o.  (/) ) )
4241sseq1d 3526 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w  o.  w ) 
C_  (/)  <->  ( (/)  o.  (/) )  C_  (/) ) )
431, 42rexsn 4072 . . . . . . 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 3520 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v 
C_  w  <->  (/)  C_  w
) )
4746imbi1d 317 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  C_  w  ->  w  e.  { (/) } )  <-> 
( (/)  C_  w  ->  w  e.  { (/) } ) ) )
4847ralbidv 2896 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  ~P  ( (/) 
X.  (/) ) ( v 
C_  w  ->  w  e.  { (/) } )  <->  A. w  e.  ~P  ( (/)  X.  (/) ) (
(/)  C_  w  ->  w  e.  { (/) } ) ) )
49 ineq1 3689 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( v  i^i  w )  =  ( (/)  i^i  w
) )
5049eleq1d 2526 . . . . . . . 8  |-  ( v  =  (/)  ->  ( ( v  i^i  w )  e.  { (/) }  <->  ( (/)  i^i  w
)  e.  { (/) } ) )
5150ralbidv 2896 . . . . . . 7  |-  ( v  =  (/)  ->  ( A. w  e.  { (/) }  (
v  i^i  w )  e.  { (/) }  <->  A. w  e.  { (/) }  ( (/)  i^i  w )  e.  { (/)
} ) )
52 sseq2 3521 . . . . . . . 8  |-  ( v  =  (/)  ->  ( (  _I  |`  (/) )  C_  v 
<->  (  _I  |`  (/) )  C_  (/) ) )
53 cnveq 5186 . . . . . . . . 9  |-  ( v  =  (/)  ->  `' v  =  `' (/) )
5453eleq1d 2526 . . . . . . . 8  |-  ( v  =  (/)  ->  ( `' v  e.  { (/) }  <->  `' (/)  e.  { (/) } ) )
55 sseq2 3521 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ( w  o.  w ) 
C_  v  <->  ( w  o.  w )  C_  (/) ) )
5655rexbidv 2968 . . . . . . . 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 4071 . . . . 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 20831 . . . . 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 4176 . . 3  |-  ( {
(/) }  e.  (UnifOn `  (/) )  ->  { { (/)
} }  C_  (UnifOn `  (/) ) )
6563, 64ax-mp 5 . 2  |-  { { (/)
} }  C_  (UnifOn `  (/) )
6616, 65eqssi 3515 1  |-  (UnifOn `  (/) )  =  { { (/)
} }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032    _I cid 4799    X. cxp 5006   `'ccnv 5007    |` cres 5010    o. ccom 5012   ` cfv 5594  UnifOncust 20827
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-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20828
This theorem is referenced by:  isusp  20889
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