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Theorem unisn3 4704
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 3865 . . 3  |-  ( A  e.  B  ->  { x  e.  B  |  x  =  A }  =  { A } )
21unieqd 4018 . 2  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  U. { A }
)
3 unisng 4024 . 2  |-  ( A  e.  B  ->  U. { A }  =  A
)
42, 3eqtrd 2467 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701   {csn 3806   U.cuni 4007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008
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