Theorem unisn 3907

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 3681	. . 3 |- { A } = { A , A }
2	1	unieqi 3901	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3905	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3348	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2254	1 |- U. { A } = A

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2800   u. cun 3196   {csn 3667   {cpr 3668   U.cuni 3891

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-uni 3892

This theorem is referenced by:  unisng  3908  uniintsnr  3962  unisuc  4508  op1sta  5216  op2nda  5219  elxp4  5222  uniabio  5295  iotass  5302  en1bg  6969  zrhval2  14623