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Theorem unisn3 4270
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 3509 . . 3  |-  ( A  e.  B  ->  { x  e.  B  |  x  =  A }  =  { A } )
21unieqd 3664 . 2  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  U. { A }
)
3 unisng 3670 . 2  |-  ( A  e.  B  ->  U. { A }  =  A
)
42, 3eqtrd 2120 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  =  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {crab 2363   {csn 3446   U.cuni 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-rab 2368  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-uni 3654
This theorem is referenced by: (None)
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