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Theorem uniintsn 4263
Description: Two ways to express " A is a singleton." See also en1 7654, en1b 7655, card1 8420, and eusn 4039. (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 3730 . . . . . 6  |-  _V  =/=  (/)
2 inteq 4229 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 4240 . . . . . . . . . . 11  |-  |^| (/)  =  _V
42, 3syl6eq 2521 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  _V )
54adantl 473 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  _V )
6 unieq 4198 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
7 uni0 4217 . . . . . . . . . . . 12  |-  U. (/)  =  (/)
86, 7syl6eq 2521 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  U. A  =  (/) )
9 eqeq1 2475 . . . . . . . . . . 11  |-  ( U. A  =  |^| A  -> 
( U. A  =  (/) 
<-> 
|^| A  =  (/) ) )
108, 9syl5ib 227 . . . . . . . . . 10  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  |^| A  =  (/) ) )
1110imp 436 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  (/) )
125, 11eqtr3d 2507 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  _V  =  (/) )
1312ex 441 . . . . . . 7  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  _V  =  (/) ) )
1413necon3d 2664 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( _V  =/=  (/)  ->  A  =/=  (/) ) )
151, 14mpi 20 . . . . 5  |-  ( U. A  =  |^| A  ->  A  =/=  (/) )
16 n0 3732 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
1715, 16sylib 201 . . . 4  |-  ( U. A  =  |^| A  ->  E. x  x  e.  A )
18 vex 3034 . . . . . . 7  |-  x  e. 
_V
19 vex 3034 . . . . . . 7  |-  y  e. 
_V
2018, 19prss 4117 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  <->  { x ,  y } 
C_  A )
21 uniss 4211 . . . . . . . . . . . . 13  |-  ( { x ,  y } 
C_  A  ->  U. {
x ,  y } 
C_  U. A )
2221adantl 473 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  U. A )
23 simpl 464 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. A  = 
|^| A )
2422, 23sseqtrd 3454 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^| A )
25 intss 4247 . . . . . . . . . . . 12  |-  ( { x ,  y } 
C_  A  ->  |^| A  C_ 
|^| { x ,  y } )
2625adantl 473 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  |^| A  C_  |^|
{ x ,  y } )
2724, 26sstrd 3428 . . . . . . . . . 10  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^|
{ x ,  y } )
2818, 19unipr 4203 . . . . . . . . . 10  |-  U. {
x ,  y }  =  ( x  u.  y )
2918, 19intpr 4259 . . . . . . . . . 10  |-  |^| { x ,  y }  =  ( x  i^i  y
)
3027, 28, 293sstr3g 3458 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( x  u.  y )  C_  (
x  i^i  y )
)
31 inss1 3643 . . . . . . . . . 10  |-  ( x  i^i  y )  C_  x
32 ssun1 3588 . . . . . . . . . 10  |-  x  C_  ( x  u.  y
)
3331, 32sstri 3427 . . . . . . . . 9  |-  ( x  i^i  y )  C_  ( x  u.  y
)
3430, 33jctir 547 . . . . . . . 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 3433 . . . . . . . . 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 3685 . . . . . . . . 9  |-  ( ( x  u.  y )  =  ( x  i^i  y )  <->  x  =  y )
3735, 36bitr3i 259 . . . . . . . 8  |-  ( ( ( x  u.  y
)  C_  ( x  i^i  y )  /\  (
x  i^i  y )  C_  ( x  u.  y
) )  <->  x  =  y )
3834, 37sylib 201 . . . . . . 7  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  x  =  y )
3938ex 441 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( { x ,  y }  C_  A  ->  x  =  y ) )
4020, 39syl5bi 225 . . . . 5  |-  ( U. A  =  |^| A  -> 
( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )
4140alrimivv 1782 . . . 4  |-  ( U. A  =  |^| A  ->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  x  =  y ) )
4217, 41jca 541 . . 3  |-  ( U. A  =  |^| A  -> 
( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
43 euabsn 4035 . . . 4  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
44 eleq1 2537 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
4544eu4 2367 . . . 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 2593 . . . . . 6  |-  { x  |  x  e.  A }  =  A
4746eqeq1i 2476 . . . . 5  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
4847exbii 1726 . . . 4  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
4943, 45, 483bitr3i 283 . . 3  |-  ( ( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )  <->  E. x  A  =  { x } )
5042, 49sylib 201 . 2  |-  ( U. A  =  |^| A  ->  E. x  A  =  { x } )
5118unisn 4205 . . . 4  |-  U. {
x }  =  x
52 unieq 4198 . . . 4  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
53 inteq 4229 . . . . 5  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
5418intsn 4262 . . . . 5  |-  |^| { x }  =  x
5553, 54syl6eq 2521 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  x )
5651, 52, 553eqtr4a 2531 . . 3  |-  ( A  =  { x }  ->  U. A  =  |^| A )
5756exlimiv 1784 . 2  |-  ( E. x  A  =  {
x }  ->  U. A  =  |^| A )
5850, 57impbii 192 1  |-  ( U. A  =  |^| A  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   {cab 2457    =/= wne 2641   _Vcvv 3031    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190   |^|cint 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-pr 3962  df-uni 4191  df-int 4227
This theorem is referenced by:  uniintab  4264
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