Theorem unisn 3930

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 3703	. . 3 |- { A } = { A , A }
2	1	unieqi 3924	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3928	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3362	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2257	1 |- U. { A } = A

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2813   u. cun 3209   {csn 3689   {cpr 3690   U.cuni 3914

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-uni 3915

This theorem is referenced by:  unisng  3931  uniintsnr  3985  unisuc  4534  op1sta  5244  op2nda  5247  elxp4  5250  uniabio  5323  iotass  5330  en1bg  7040  zrhval2  14767