Theorem unisn 3880

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 3657	. . 3 |- { A } = { A , A }
2	1	unieqi 3874	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3878	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3324	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2232	1 |- U. { A } = A

Syntax hints:   = wceq 1373   e. wcel 2178   _Vcvv 2776   u. cun 3172   {csn 3643   {cpr 3644   U.cuni 3864

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865

This theorem is referenced by:  unisng  3881  uniintsnr  3935  unisuc  4478  op1sta  5183  op2nda  5186  elxp4  5189  uniabio  5261  iotass  5268  en1bg  6915  zrhval2  14496