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Theorem unisn 3866
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 3647 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3860 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3864 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3316 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2230 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1373    e. wcel 2176   _Vcvv 2772    u. cun 3164   {csn 3633   {cpr 3634   U.cuni 3850
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851
This theorem is referenced by:  unisng  3867  uniintsnr  3921  unisuc  4460  op1sta  5164  op2nda  5167  elxp4  5170  uniabio  5242  iotass  5249  en1bg  6892  zrhval2  14381
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