Theorem unisn 3909

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 3683	. . 3 |- { A } = { A , A }
2	1	unieqi 3903	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3907	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3350	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2256	1 |- U. { A } = A

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   {csn 3669   {cpr 3670   U.cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894

This theorem is referenced by:  unisng  3910  uniintsnr  3964  unisuc  4510  op1sta  5218  op2nda  5221  elxp4  5224  uniabio  5297  iotass  5304  en1bg  6973  zrhval2  14632