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Theorem unisn 3570
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 V
Assertion
Ref Expression
unisn {A} = A

Proof of Theorem unisn
StepHypRef Expression
1 dfsn2 3364 . . 3 {A} = {A, A}
21unieqi 3564 . 2 {A} = {A, A}
3 unisn.1 . . 3 A V
43, 3unipr 3568 . 2 {A, A} = (AA)
5 unidm 3063 . 2 (AA) = A
62, 4, 53eqtri 2046 1 {A} = A
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  cun 2892  {csn 3350  {cpr 3351   cuni 3554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by:  unisng  3571  uniintsnr  3625  unisuc  4099  op1sta  4729  op2nda  4732  elxp4  4735  uniabio  4804  iotass  4811
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