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Theorem uniintsn 4250
Description: Two ways to express " A is a singleton." See also en1 7519, en1b 7520, card1 8280, and eusn 4033. (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 3732 . . . . . 6  |-  _V  =/=  (/)
2 inteq 4215 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 4226 . . . . . . . . . . 11  |-  |^| (/)  =  _V
42, 3syl6eq 2449 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  _V )
54adantl 464 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  _V )
6 unieq 4184 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
7 uni0 4203 . . . . . . . . . . . 12  |-  U. (/)  =  (/)
86, 7syl6eq 2449 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  U. A  =  (/) )
9 eqeq1 2396 . . . . . . . . . . 11  |-  ( U. A  =  |^| A  -> 
( U. A  =  (/) 
<-> 
|^| A  =  (/) ) )
108, 9syl5ib 219 . . . . . . . . . 10  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  |^| A  =  (/) ) )
1110imp 427 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  (/) )
125, 11eqtr3d 2435 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  _V  =  (/) )
1312ex 432 . . . . . . 7  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  _V  =  (/) ) )
1413necon3d 2616 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( _V  =/=  (/)  ->  A  =/=  (/) ) )
151, 14mpi 17 . . . . 5  |-  ( U. A  =  |^| A  ->  A  =/=  (/) )
16 n0 3734 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
1715, 16sylib 196 . . . 4  |-  ( U. A  =  |^| A  ->  E. x  x  e.  A )
18 vex 3050 . . . . . . 7  |-  x  e. 
_V
19 vex 3050 . . . . . . 7  |-  y  e. 
_V
2018, 19prss 4111 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  <->  { x ,  y } 
C_  A )
21 uniss 4197 . . . . . . . . . . . . 13  |-  ( { x ,  y } 
C_  A  ->  U. {
x ,  y } 
C_  U. A )
2221adantl 464 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  U. A )
23 simpl 455 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. A  = 
|^| A )
2422, 23sseqtrd 3466 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^| A )
25 intss 4233 . . . . . . . . . . . 12  |-  ( { x ,  y } 
C_  A  ->  |^| A  C_ 
|^| { x ,  y } )
2625adantl 464 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  |^| A  C_  |^|
{ x ,  y } )
2724, 26sstrd 3440 . . . . . . . . . 10  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^|
{ x ,  y } )
2818, 19unipr 4189 . . . . . . . . . 10  |-  U. {
x ,  y }  =  ( x  u.  y )
2918, 19intpr 4246 . . . . . . . . . 10  |-  |^| { x ,  y }  =  ( x  i^i  y
)
3027, 28, 293sstr3g 3470 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( x  u.  y )  C_  (
x  i^i  y )
)
31 inss1 3645 . . . . . . . . . 10  |-  ( x  i^i  y )  C_  x
32 ssun1 3594 . . . . . . . . . 10  |-  x  C_  ( x  u.  y
)
3331, 32sstri 3439 . . . . . . . . 9  |-  ( x  i^i  y )  C_  ( x  u.  y
)
3430, 33jctir 536 . . . . . . . 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 3445 . . . . . . . . 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 3687 . . . . . . . . 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 432 . . . . . 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 1735 . . . 4  |-  ( U. A  =  |^| A  ->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  x  =  y ) )
4217, 41jca 530 . . 3  |-  ( U. A  =  |^| A  -> 
( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
43 euabsn 4029 . . . 4  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
44 eleq1 2464 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
4544eu4 2280 . . . 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 2532 . . . . . 6  |-  { x  |  x  e.  A }  =  A
4746eqeq1i 2399 . . . . 5  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
4847exbii 1682 . . . 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 4191 . . . 4  |-  U. {
x }  =  x
52 unieq 4184 . . . 4  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
53 inteq 4215 . . . . 5  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
5418intsn 4249 . . . . 5  |-  |^| { x }  =  x
5553, 54syl6eq 2449 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  x )
5651, 52, 553eqtr4a 2459 . . 3  |-  ( A  =  { x }  ->  U. A  =  |^| A )
5756exlimiv 1737 . 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 367   A.wal 1397    = wceq 1399   E.wex 1627    e. wcel 1836   E!weu 2228   {cab 2377    =/= wne 2587   _Vcvv 3047    u. cun 3400    i^i cin 3401    C_ wss 3402   (/)c0 3724   {csn 3957   {cpr 3959   U.cuni 4176   |^|cint 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-sn 3958  df-pr 3960  df-uni 4177  df-int 4213
This theorem is referenced by:  uniintab  4251
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