Theorem unisn 3856

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

Step	Hyp	Ref	Expression
1		dfsn2 3637	. . 3 |- { A } = { A , A }
2	1	unieqi 3850	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3854	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3307	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2221	1 |- U. { A } = A

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   u. cun 3155   {csn 3623   {cpr 3624   U.cuni 3840

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-uni 3841

This theorem is referenced by:  unisng  3857  uniintsnr  3911  unisuc  4449  op1sta  5152  op2nda  5155  elxp4  5158  uniabio  5230  iotass  5237  en1bg  6868  zrhval2  14251