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Theorem uniintsn 4275
Description: Two ways to express " A is a singleton." See also en1 7641, en1b 7642, card1 8407, and eusn 4051. (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 3741 . . . . . 6  |-  _V  =/=  (/)
2 inteq 4240 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 4251 . . . . . . . . . . 11  |-  |^| (/)  =  _V
42, 3syl6eq 2503 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  _V )
54adantl 468 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  _V )
6 unieq 4209 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
7 uni0 4228 . . . . . . . . . . . 12  |-  U. (/)  =  (/)
86, 7syl6eq 2503 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  U. A  =  (/) )
9 eqeq1 2457 . . . . . . . . . . 11  |-  ( U. A  =  |^| A  -> 
( U. A  =  (/) 
<-> 
|^| A  =  (/) ) )
108, 9syl5ib 223 . . . . . . . . . 10  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  |^| A  =  (/) ) )
1110imp 431 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  |^| A  =  (/) )
125, 11eqtr3d 2489 . . . . . . . 8  |-  ( ( U. A  =  |^| A  /\  A  =  (/) )  ->  _V  =  (/) )
1312ex 436 . . . . . . 7  |-  ( U. A  =  |^| A  -> 
( A  =  (/)  ->  _V  =  (/) ) )
1413necon3d 2647 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( _V  =/=  (/)  ->  A  =/=  (/) ) )
151, 14mpi 20 . . . . 5  |-  ( U. A  =  |^| A  ->  A  =/=  (/) )
16 n0 3743 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
1715, 16sylib 200 . . . 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 4129 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  <->  { x ,  y } 
C_  A )
21 uniss 4222 . . . . . . . . . . . . 13  |-  ( { x ,  y } 
C_  A  ->  U. {
x ,  y } 
C_  U. A )
2221adantl 468 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  U. A )
23 simpl 459 . . . . . . . . . . . 12  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. A  = 
|^| A )
2422, 23sseqtrd 3470 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^| A )
25 intss 4258 . . . . . . . . . . . 12  |-  ( { x ,  y } 
C_  A  ->  |^| A  C_ 
|^| { x ,  y } )
2625adantl 468 . . . . . . . . . . 11  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  |^| A  C_  |^|
{ x ,  y } )
2724, 26sstrd 3444 . . . . . . . . . 10  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  U. { x ,  y }  C_  |^|
{ x ,  y } )
2818, 19unipr 4214 . . . . . . . . . 10  |-  U. {
x ,  y }  =  ( x  u.  y )
2918, 19intpr 4271 . . . . . . . . . 10  |-  |^| { x ,  y }  =  ( x  i^i  y
)
3027, 28, 293sstr3g 3474 . . . . . . . . 9  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  ( x  u.  y )  C_  (
x  i^i  y )
)
31 inss1 3654 . . . . . . . . . 10  |-  ( x  i^i  y )  C_  x
32 ssun1 3599 . . . . . . . . . 10  |-  x  C_  ( x  u.  y
)
3331, 32sstri 3443 . . . . . . . . 9  |-  ( x  i^i  y )  C_  ( x  u.  y
)
3430, 33jctir 541 . . . . . . . 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 3449 . . . . . . . . 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 3696 . . . . . . . . 9  |-  ( ( x  u.  y )  =  ( x  i^i  y )  <->  x  =  y )
3735, 36bitr3i 255 . . . . . . . 8  |-  ( ( ( x  u.  y
)  C_  ( x  i^i  y )  /\  (
x  i^i  y )  C_  ( x  u.  y
) )  <->  x  =  y )
3834, 37sylib 200 . . . . . . 7  |-  ( ( U. A  =  |^| A  /\  { x ,  y }  C_  A
)  ->  x  =  y )
3938ex 436 . . . . . 6  |-  ( U. A  =  |^| A  -> 
( { x ,  y }  C_  A  ->  x  =  y ) )
4020, 39syl5bi 221 . . . . 5  |-  ( U. A  =  |^| A  -> 
( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )
4140alrimivv 1776 . . . 4  |-  ( U. A  =  |^| A  ->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  x  =  y ) )
4217, 41jca 535 . . 3  |-  ( U. A  =  |^| A  -> 
( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) ) )
43 euabsn 4047 . . . 4  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
44 eleq1 2519 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
4544eu4 2349 . . . 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 2575 . . . . . 6  |-  { x  |  x  e.  A }  =  A
4746eqeq1i 2458 . . . . 5  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
4847exbii 1720 . . . 4  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
4943, 45, 483bitr3i 279 . . 3  |-  ( ( E. x  x  e.  A  /\  A. x A. y ( ( x  e.  A  /\  y  e.  A )  ->  x  =  y ) )  <->  E. x  A  =  { x } )
5042, 49sylib 200 . 2  |-  ( U. A  =  |^| A  ->  E. x  A  =  { x } )
5118unisn 4216 . . . 4  |-  U. {
x }  =  x
52 unieq 4209 . . . 4  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
53 inteq 4240 . . . . 5  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
5418intsn 4274 . . . . 5  |-  |^| { x }  =  x
5553, 54syl6eq 2503 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  x )
5651, 52, 553eqtr4a 2513 . . 3  |-  ( A  =  { x }  ->  U. A  =  |^| A )
5756exlimiv 1778 . 2  |-  ( E. x  A  =  {
x }  ->  U. A  =  |^| A )
5850, 57impbii 191 1  |-  ( U. A  =  |^| A  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889   E!weu 2301   {cab 2439    =/= wne 2624   _Vcvv 3047    u. cun 3404    i^i cin 3405    C_ wss 3406   (/)c0 3733   {csn 3970   {cpr 3972   U.cuni 4201   |^|cint 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-sn 3971  df-pr 3973  df-uni 4202  df-int 4238
This theorem is referenced by:  uniintab  4276
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