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Theorem unisngl 20479
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 4166 . 2  |-  U. C  =  U. { u  |  E. x  e.  X  u  =  { x } }
3 simpl 458 . . . . . . . . 9  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  y  e.  u )
4 simpr 462 . . . . . . . . 9  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  u  =  { x } )
53, 4eleqtrd 2503 . . . . . . . 8  |-  ( ( y  e.  u  /\  u  =  { x } )  ->  y  e.  { x } )
65exlimiv 1770 . . . . . . 7  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  ->  y  e.  {
x } )
7 eqid 2423 . . . . . . . 8  |-  { x }  =  { x }
8 snex 4600 . . . . . . . . 9  |-  { x }  e.  _V
9 eleq2 2490 . . . . . . . . . 10  |-  ( u  =  { x }  ->  ( y  e.  u  <->  y  e.  { x }
) )
10 eqeq1 2427 . . . . . . . . . 10  |-  ( u  =  { x }  ->  ( u  =  {
x }  <->  { x }  =  { x } ) )
119, 10anbi12d 715 . . . . . . . . 9  |-  ( u  =  { x }  ->  ( ( y  e.  u  /\  u  =  { x } )  <-> 
( y  e.  {
x }  /\  {
x }  =  {
x } ) ) )
128, 11spcev 3111 . . . . . . . 8  |-  ( ( y  e.  { x }  /\  { x }  =  { x } )  ->  E. u ( y  e.  u  /\  u  =  { x } ) )
137, 12mpan2 675 . . . . . . 7  |-  ( y  e.  { x }  ->  E. u ( y  e.  u  /\  u  =  { x } ) )
146, 13impbii 190 . . . . . 6  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  <-> 
y  e.  { x } )
15 elsn 3950 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
16 equcom 1848 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
1714, 15, 163bitri 274 . . . . 5  |-  ( E. u ( y  e.  u  /\  u  =  { x } )  <-> 
x  =  y )
1817rexbii 2861 . . . 4  |-  ( E. x  e.  X  E. u ( y  e.  u  /\  u  =  { x } )  <->  E. x  e.  X  x  =  y )
19 r19.42v 2917 . . . . . 6  |-  ( E. x  e.  X  ( y  e.  u  /\  u  =  { x } )  <->  ( y  e.  u  /\  E. x  e.  X  u  =  { x } ) )
2019exbii 1712 . . . . 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 3038 . . . . 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 4168 . . . . 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 281 . . . 4  |-  ( y  e.  U. { u  |  E. x  e.  X  u  =  { x } }  <->  E. x  e.  X  E. u ( y  e.  u  /\  u  =  { x } ) )
24 risset 2887 . . . 4  |-  ( y  e.  X  <->  E. x  e.  X  x  =  y )
2518, 23, 243bitr4i 280 . . 3  |-  ( y  e.  U. { u  |  E. x  e.  X  u  =  { x } }  <->  y  e.  X
)
2625eqriv 2420 . 2  |-  U. {
u  |  E. x  e.  X  u  =  { x } }  =  X
272, 26eqtr2i 2446 1  |-  X  = 
U. C
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2409   E.wrex 2710   {csn 3936   U.cuni 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-v 3019  df-dif 3377  df-un 3379  df-nul 3700  df-sn 3937  df-pr 3939  df-uni 4158
This theorem is referenced by:  dissnref  20480  dissnlocfin  20481
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