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Theorem unisn 3722
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisn.1  |-  A  e. 
_V
Assertion
Ref Expression
unisn  |-  U. { A }  =  A

Proof of Theorem unisn
StepHypRef Expression
1 dfsn2 3511 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3716 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3720 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3189 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2142 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   _Vcvv 2660    u. cun 3039   {csn 3497   {cpr 3498   U.cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707
This theorem is referenced by:  unisng  3723  uniintsnr  3777  unisuc  4305  op1sta  4990  op2nda  4993  elxp4  4996  uniabio  5068  iotass  5075  en1bg  6662
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