Theorem unisn 3805

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 3590	. . 3 |- { A } = { A , A }
2	1	unieqi 3799	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3803	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3265	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2190	1 |- U. { A } = A

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   {csn 3576   {cpr 3577   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by:  unisng  3806  uniintsnr  3860  unisuc  4391  op1sta  5085  op2nda  5088  elxp4  5091  uniabio  5163  iotass  5170  en1bg  6766