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Theorem uniintsn 4162
Description: Two ways to express " A is a singleton." See also en1 7372, en1b 7373, card1 8134, and eusn 3948. (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 3641 . . . . . 6  |-  _V  =/=  (/)
2 inteq 4128 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 4139 . . . . . . . . . . 11  |-  |^| (/)  =  _V
42, 3syl6eq 2489 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  _V )
54adantl 463 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  _V )
6 unieq 4096 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
7 uni0 4115 . . . . . . . . . . . 12  |-  U. (/)  =  (/)
86, 7syl6eq 2489 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  U. A  =  (/) )
9 eqeq1 2447 . . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  _V  =  (/) )
1312ex 434 . . . . . . 7  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  _V  =  (/) ) )
1413necon3d 2644 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( _V  =/=  (/)  ->  A  =/=  (/) ) )
151, 14mpi 17 . . . . 5  |-  ( U. A  =  |^| A  ->  A  =/=  (/) )
16 n0 3643 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
1715, 16sylib 196 . . . 4  |-  ( U. A  =  |^| A  ->  E. x  x  e.  A )
18 vex 2973 . . . . . . 7  |-  x  e. 
_V
19 vex 2973 . . . . . . 7  |-  y  e. 
_V
2018, 19prss 4024 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  <->  { x ,  y } 
C_  A )
21 uniss 4109 . . . . . . . . . . . . 13  |-  ( { x ,  y } 
C_  A  ->  U. {
x ,  y } 
C_  U. A )
2221adantl 463 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  U. A )
23 simpl 454 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. A  = 
|^| A )
2422, 23sseqtrd 3389 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^| A )
25 intss 4146 . . . . . . . . . . . 12  |-  ( { x ,  y } 
C_  A  ->  |^| A  C_ 
|^| { x ,  y } )
2625adantl 463 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  |^| A  C_  |^|
{ x ,  y } )
2724, 26sstrd 3363 . . . . . . . . . 10  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^|
{ x ,  y } )
2818, 19unipr 4101 . . . . . . . . . 10  |-  U. {
x ,  y }  =  ( x  u.  y )
2918, 19intpr 4158 . . . . . . . . . 10  |-  |^| { x ,  y }  =  ( x  i^i  y
)
3027, 28, 293sstr3g 3393 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( x  u.  y )  C_  (
x  i^i  y )
)
31 inss1 3567 . . . . . . . . . 10  |-  ( x  i^i  y )  C_  x
32 ssun1 3516 . . . . . . . . . 10  |-  x  C_  ( x  u.  y
)
3331, 32sstri 3362 . . . . . . . . 9  |-  ( x  i^i  y )  C_  ( x  u.  y
)
3430, 33jctir 535 . . . . . . . 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 3368 . . . . . . . . 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 3598 . . . . . . . . 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 1691 . . . 4  |-  ( U. A  =  |^| A  ->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  x  =  y ) )
4217, 41jca 529 . . 3  |-  ( U. A  =  |^| A  -> 
( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
43 euabsn 3944 . . . 4  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
44 eleq1 2501 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
4544eu4 2321 . . . 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 2558 . . . . . 6  |-  { x  |  x  e.  A }  =  A
4746eqeq1i 2448 . . . . 5  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
4847exbii 1639 . . . 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 4103 . . . 4  |-  U. {
x }  =  x
52 unieq 4096 . . . 4  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
53 inteq 4128 . . . . 5  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
5418intsn 4161 . . . . 5  |-  |^| { x }  =  x
5553, 54syl6eq 2489 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  x )
5651, 52, 553eqtr4a 2499 . . 3  |-  ( A  =  { x }  ->  U. A  =  |^| A )
5756exlimiv 1693 . 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 1362    = wceq 1364   E.wex 1591    e. wcel 1761   E!weu 2257   {cab 2427    =/= wne 2604   _Vcvv 2970    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   {cpr 3876   U.cuni 4088   |^|cint 4125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-sn 3875  df-pr 3877  df-uni 4089  df-int 4126
This theorem is referenced by:  uniintab  4163  reusv6OLD  4500  reusv7OLD  4501
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