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Theorem unisn3 4523
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 3697 . . 3  |-  ( A  e.  B  ->  { x  e.  B  |  x  =  A }  =  { A } )
21unieqd 3838 . 2  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  U. { A }
)
3 unisng 3844 . 2  |-  ( A  e.  B  ->  U. { A }  =  A
)
42, 3eqtrd 2315 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   {csn 3640   U.cuni 3827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828
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