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Theorem unisng 3803
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
Assertion
Ref Expression
unisng  |-  ( A  e.  V  ->  U. { A }  =  A
)

Proof of Theorem unisng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3584 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21unieqd 3797 . . 3  |-  ( x  =  A  ->  U. {
x }  =  U. { A } )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2179 . 2  |-  ( x  =  A  ->  ( U. { x }  =  x 
<-> 
U. { A }  =  A ) )
5 vex 2727 . . 3  |-  x  e. 
_V
65unisn 3802 . 2  |-  U. {
x }  =  x
74, 6vtoclg 2784 1  |-  ( A  e.  V  ->  U. { A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1342    e. wcel 2135   {csn 3573   U.cuni 3786
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-uni 3787
This theorem is referenced by:  dfnfc2  3804  unisucg  4389  unisn3  4420  opswapg  5087  funfvdm  5546  en2other2  7146
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