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Theorem unisn 3825
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 3606 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3819 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3823 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3278 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2202 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127   {csn 3592   {cpr 3593   U.cuni 3809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-uni 3810
This theorem is referenced by:  unisng  3826  uniintsnr  3880  unisuc  4413  op1sta  5110  op2nda  5113  elxp4  5116  uniabio  5188  iotass  5195  en1bg  6799
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