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Theorem unisn 3812
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 3597 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3806 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3810 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3270 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2195 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141   _Vcvv 2730    u. cun 3119   {csn 3583   {cpr 3584   U.cuni 3796
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797
This theorem is referenced by:  unisng  3813  uniintsnr  3867  unisuc  4398  op1sta  5092  op2nda  5095  elxp4  5098  uniabio  5170  iotass  5177  en1bg  6778
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