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Theorem unisn 3843
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 3624 . . 3 {𝐴} = {𝐴, 𝐴}
21unieqi 3837 . 2 {𝐴} = {𝐴, 𝐴}
3 unisn.1 . . 3 𝐴 ∈ V
43, 3unipr 3841 . 2 {𝐴, 𝐴} = (𝐴𝐴)
5 unidm 3293 . 2 (𝐴𝐴) = 𝐴
62, 4, 53eqtri 2214 1 {𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  {csn 3610  {cpr 3611   cuni 3827
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3616  df-pr 3617  df-uni 3828
This theorem is referenced by:  unisng  3844  uniintsnr  3898  unisuc  4434  op1sta  5131  op2nda  5134  elxp4  5137  uniabio  5209  iotass  5216  en1bg  6830  zrhval2  13941
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