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Theorem uniintsn 4165
Description: Two ways to express " A is a singleton." See also en1 7376, en1b 7377, card1 8138, and eusn 3951. (Contributed by NM, 2-Aug-2010.)
Assertion
Ref Expression
uniintsn  |-  ( U. A  =  |^| A  <->  E. x  A  =  { x } )
Distinct variable group:    x, A

Proof of Theorem uniintsn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vn0 3644 . . . . . 6  |-  _V  =/=  (/)
2 inteq 4131 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 4142 . . . . . . . . . . 11  |-  |^| (/)  =  _V
42, 3syl6eq 2491 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  _V )
54adantl 466 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  _V )
6 unieq 4099 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
7 uni0 4118 . . . . . . . . . . . 12  |-  U. (/)  =  (/)
86, 7syl6eq 2491 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  U. A  =  (/) )
9 eqeq1 2449 . . . . . . . . . . 11  |-  ( U. A  =  |^| A  -> 
( U. A  =  (/) 
<-> 
|^| A  =  (/) ) )
108, 9syl5ib 219 . . . . . . . . . 10  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  |^| A  =  (/) ) )
1110imp 429 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  (/) )
125, 11eqtr3d 2477 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  _V  =  (/) )
1312ex 434 . . . . . . 7  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  _V  =  (/) ) )
1413necon3d 2646 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( _V  =/=  (/)  ->  A  =/=  (/) ) )
151, 14mpi 17 . . . . 5  |-  ( U. A  =  |^| A  ->  A  =/=  (/) )
16 n0 3646 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
1715, 16sylib 196 . . . 4  |-  ( U. A  =  |^| A  ->  E. x  x  e.  A )
18 vex 2975 . . . . . . 7  |-  x  e. 
_V
19 vex 2975 . . . . . . 7  |-  y  e. 
_V
2018, 19prss 4027 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  <->  { x ,  y } 
C_  A )
21 uniss 4112 . . . . . . . . . . . . 13  |-  ( { x ,  y } 
C_  A  ->  U. {
x ,  y } 
C_  U. A )
2221adantl 466 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  U. A )
23 simpl 457 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. A  = 
|^| A )
2422, 23sseqtrd 3392 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^| A )
25 intss 4149 . . . . . . . . . . . 12  |-  ( { x ,  y } 
C_  A  ->  |^| A  C_ 
|^| { x ,  y } )
2625adantl 466 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  |^| A  C_  |^|
{ x ,  y } )
2724, 26sstrd 3366 . . . . . . . . . 10  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^|
{ x ,  y } )
2818, 19unipr 4104 . . . . . . . . . 10  |-  U. {
x ,  y }  =  ( x  u.  y )
2918, 19intpr 4161 . . . . . . . . . 10  |-  |^| { x ,  y }  =  ( x  i^i  y
)
3027, 28, 293sstr3g 3396 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( x  u.  y )  C_  (
x  i^i  y )
)
31 inss1 3570 . . . . . . . . . 10  |-  ( x  i^i  y )  C_  x
32 ssun1 3519 . . . . . . . . . 10  |-  x  C_  ( x  u.  y
)
3331, 32sstri 3365 . . . . . . . . 9  |-  ( x  i^i  y )  C_  ( x  u.  y
)
3430, 33jctir 538 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( (
x  u.  y ) 
C_  ( x  i^i  y )  /\  (
x  i^i  y )  C_  ( x  u.  y
) ) )
35 eqss 3371 . . . . . . . . 9  |-  ( ( x  u.  y )  =  ( x  i^i  y )  <->  ( (
x  u.  y ) 
C_  ( x  i^i  y )  /\  (
x  i^i  y )  C_  ( x  u.  y
) ) )
36 uneqin 3601 . . . . . . . . 9  |-  ( ( x  u.  y )  =  ( x  i^i  y )  <->  x  =  y )
3735, 36bitr3i 251 . . . . . . . 8  |-  ( ( ( x  u.  y
)  C_  ( x  i^i  y )  /\  (
x  i^i  y )  C_  ( x  u.  y
) )  <->  x  =  y )
3834, 37sylib 196 . . . . . . 7  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  x  =  y )
3938ex 434 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( { x ,  y }  C_  A  ->  x  =  y ) )
4020, 39syl5bi 217 . . . . 5  |-  ( U. A  =  |^| A  -> 
( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )
4140alrimivv 1686 . . . 4  |-  ( U. A  =  |^| A  ->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  x  =  y ) )
4217, 41jca 532 . . 3  |-  ( U. A  =  |^| A  -> 
( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
43 euabsn 3947 . . . 4  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
44 eleq1 2503 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
4544eu4 2318 . . . 4  |-  ( E! x  x  e.  A  <->  ( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
46 abid2 2560 . . . . . 6  |-  { x  |  x  e.  A }  =  A
4746eqeq1i 2450 . . . . 5  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
4847exbii 1634 . . . 4  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
4943, 45, 483bitr3i 275 . . 3  |-  ( ( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )  <->  E. x  A  =  { x } )
5042, 49sylib 196 . 2  |-  ( U. A  =  |^| A  ->  E. x  A  =  { x } )
5118unisn 4106 . . . 4  |-  U. {
x }  =  x
52 unieq 4099 . . . 4  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
53 inteq 4131 . . . . 5  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
5418intsn 4164 . . . . 5  |-  |^| { x }  =  x
5553, 54syl6eq 2491 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  x )
5651, 52, 553eqtr4a 2501 . . 3  |-  ( A  =  { x }  ->  U. A  =  |^| A )
5756exlimiv 1688 . 2  |-  ( E. x  A  =  {
x }  ->  U. A  =  |^| A )
5850, 57impbii 188 1  |-  ( U. A  =  |^| A  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   {cab 2429    =/= wne 2606   _Vcvv 2972    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879   U.cuni 4091   |^|cint 4128
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-sn 3878  df-pr 3880  df-uni 4092  df-int 4129
This theorem is referenced by:  uniintab  4166  reusv6OLD  4503  reusv7OLD  4504
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