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Theorem unisngl 20197
Description: Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.)
Hypothesis
Ref Expression
dissnref.c  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
Assertion
Ref Expression
unisngl  |-  X  = 
U. C
Distinct variable groups:    u, C, x    u, X, x

Proof of Theorem unisngl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dissnref.c . . 3  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
21unieqi 4244 . 2  |-  U. C  =  U. { u  |  E. x  e.  X  u  =  { x } }
3 simpl 455 . . . . . . . . 9  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  y  e.  u )
4 simpr 459 . . . . . . . . 9  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  u  =  { x } )
53, 4eleqtrd 2544 . . . . . . . 8  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  y  e.  { x } )
65exlimiv 1727 . . . . . . 7  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  ->  y  e.  {
x } )
7 eqid 2454 . . . . . . . 8  |-  { x }  =  { x }
8 snex 4678 . . . . . . . . 9  |-  { x }  e.  _V
9 eleq2 2527 . . . . . . . . . 10  |-  ( u  =  { x }  ->  ( y  e.  u  <->  y  e.  { x }
) )
10 eqeq1 2458 . . . . . . . . . 10  |-  ( u  =  { x }  ->  ( u  =  {
x }  <->  { x }  =  { x } ) )
119, 10anbi12d 708 . . . . . . . . 9  |-  ( u  =  { x }  ->  ( ( y  e.  u  /\  u  =  { x } )  <-> 
( y  e.  {
x }  /\  {
x }  =  {
x } ) ) )
128, 11spcev 3198 . . . . . . . 8  |-  ( ( y  e.  { x }  /\  { x }  =  { x } )  ->  E. u ( y  e.  u  /\  u  =  { x } ) )
137, 12mpan2 669 . . . . . . 7  |-  ( y  e.  { x }  ->  E. u ( y  e.  u  /\  u  =  { x } ) )
146, 13impbii 188 . . . . . 6  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  <-> 
y  e.  { x } )
15 elsn 4030 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
16 equcom 1799 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
1714, 15, 163bitri 271 . . . . 5  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  <-> 
x  =  y )
1817rexbii 2956 . . . 4  |-  ( E. x  e.  X  E. u ( y  e.  u  /\  u  =  { x } )  <->  E. x  e.  X  x  =  y )
19 r19.42v 3009 . . . . . 6  |-  ( E. x  e.  X  ( y  e.  u  /\  u  =  { x } )  <->  ( y  e.  u  /\  E. x  e.  X  u  =  { x } ) )
2019exbii 1672 . . . . 5  |-  ( E. u E. x  e.  X  ( y  e.  u  /\  u  =  { x } )  <->  E. u ( y  e.  u  /\  E. x  e.  X  u  =  { x } ) )
21 rexcom4 3126 . . . . 5  |-  ( E. x  e.  X  E. u ( y  e.  u  /\  u  =  { x } )  <->  E. u E. x  e.  X  ( y  e.  u  /\  u  =  { x } ) )
22 eluniab 4246 . . . . 5  |-  ( y  e.  U. { u  |  E. x  e.  X  u  =  { x } }  <->  E. u ( y  e.  u  /\  E. x  e.  X  u  =  { x } ) )
2320, 21, 223bitr4ri 278 . . . 4  |-  ( y  e.  U. { u  |  E. x  e.  X  u  =  { x } }  <->  E. x  e.  X  E. u ( y  e.  u  /\  u  =  { x } ) )
24 risset 2979 . . . 4  |-  ( y  e.  X  <->  E. x  e.  X  x  =  y )
2518, 23, 243bitr4i 277 . . 3  |-  ( y  e.  U. { u  |  E. x  e.  X  u  =  { x } }  <->  y  e.  X
)
2625eqriv 2450 . 2  |-  U. {
u  |  E. x  e.  X  u  =  { x } }  =  X
272, 26eqtr2i 2484 1  |-  X  = 
U. C
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   E.wrex 2805   {csn 4016   U.cuni 4235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-nul 3784  df-sn 4017  df-pr 4019  df-uni 4236
This theorem is referenced by:  dissnref  20198  dissnlocfin  20199
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