MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisn3 Unicode version

Theorem unisn3 4539
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 3710 . . 3  |-  ( A  e.  B  ->  { x  e.  B  |  x  =  A }  =  { A } )
21unieqd 3854 . 2  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  U. { A }
)
3 unisng 3860 . 2  |-  ( A  e.  B  ->  U. { A }  =  A
)
42, 3eqtrd 2328 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   {csn 3653   U.cuni 3843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844
  Copyright terms: Public domain W3C validator