Theorem unisn 1932

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.

Hypothesis

Ref	Expression
unisn.1	|- A e. V

Assertion

Ref	Expression
unisn	|- U.{A} = A

Proof of Theorem unisn

Step	Hyp	Ref	Expression
1		dfsn2 1819	. . 3 |- {A} = {A, A}
2	1	unieqi 1928	. 2 |- U.{A} = U.{A, A}
3		unisn.1	. . 3 |- A e. V
4	3, 3	unpr 1930	. 2 |- U.{A, A} = (A u. A)
5		unidm 1603	. 2 |- (A u. A) = A
6	2, 4, 5	3eqtr 1123	1 |- U.{A} = A

Syntax hints:   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485  {csn 1808  {cpr 1809  U.cuni 1919

This theorem is referenced by:  unisng 1933  uni0 1938  unidif0 1944  euuni 1954  reucl 1957  unisuc 2299  onuninsuc 2356  op1sta 2635  fvex 2838  funfv 2862  ecqs 3233  xpcomen 3343

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-uni 1920

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