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Theorem unisn 3664
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 3455 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3658 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3662 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3141 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2112 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995   {csn 3441   {cpr 3442   U.cuni 3648
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-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649
This theorem is referenced by:  unisng  3665  uniintsnr  3719  unisuc  4231  op1sta  4899  op2nda  4902  elxp4  4905  uniabio  4977  iotass  4984  en1bg  6497
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