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Theorem unisn 3865
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 3646 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3859 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3863 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3315 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2229 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1372    e. wcel 2175   _Vcvv 2771    u. cun 3163   {csn 3632   {cpr 3633   U.cuni 3849
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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850
This theorem is referenced by:  unisng  3866  uniintsnr  3920  unisuc  4459  op1sta  5163  op2nda  5166  elxp4  5169  uniabio  5241  iotass  5248  en1bg  6891  zrhval2  14352
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