Theorem unisn 3812

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 3597	. . 3 |- { A } = { A , A }
2	1	unieqi 3806	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3810	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3270	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2195	1 |- U. { A } = A

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   {csn 3583   {cpr 3584   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  unisng  3813  uniintsnr  3867  unisuc  4398  op1sta  5092  op2nda  5095  elxp4  5098  uniabio  5170  iotass  5177  en1bg  6778