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Theorem uniintsnr 3865
Description: The union and intersection of a singleton are equal. See also eusn 3655. (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
uniintsnr  |-  ( E. x  A  =  {
x }  ->  U. A  =  |^| A )
Distinct variable group:    x, A

Proof of Theorem uniintsnr
StepHypRef Expression
1 vex 2733 . . . 4  |-  x  e. 
_V
21unisn 3810 . . 3  |-  U. {
x }  =  x
3 unieq 3803 . . 3  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
4 inteq 3832 . . . 4  |-  ( A  =  { x }  ->  |^| A  =  |^| { x } )
51intsn 3864 . . . 4  |-  |^| { x }  =  x
64, 5eqtrdi 2219 . . 3  |-  ( A  =  { x }  ->  |^| A  =  x )
72, 3, 63eqtr4a 2229 . 2  |-  ( A  =  { x }  ->  U. A  =  |^| A )
87exlimiv 1591 1  |-  ( E. x  A  =  {
x }  ->  U. A  =  |^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   E.wex 1485   {csn 3581   U.cuni 3794   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830
This theorem is referenced by:  uniintabim  3866
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