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Theorem unisn 3788
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 3574 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3782 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3786 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3250 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2182 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1335    e. wcel 2128   _Vcvv 2712    u. cun 3100   {csn 3560   {cpr 3561   U.cuni 3772
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-uni 3773
This theorem is referenced by:  unisng  3789  uniintsnr  3843  unisuc  4373  op1sta  5067  op2nda  5070  elxp4  5073  uniabio  5145  iotass  5152  en1bg  6745
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