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Theorem unisn3 4653
Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
Assertion
Ref Expression
unisn3  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem unisn3
StepHypRef Expression
1 rabsn 3817 . . 3  |-  ( A  e.  B  ->  { x  e.  B  |  x  =  A }  =  { A } )
21unieqd 3969 . 2  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  U. { A }
)
3 unisng 3975 . 2  |-  ( A  e.  B  ->  U. { A }  =  A
)
42, 3eqtrd 2420 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {crab 2654   {csn 3758   U.cuni 3958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-rab 2659  df-v 2902  df-un 3269  df-sn 3764  df-pr 3765  df-uni 3959
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