Theorem unisn 3865

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 3646	. . 3 |- { A } = { A , A }
2	1	unieqi 3859	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3863	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3315	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2229	1 |- U. { A } = A

Syntax hints:   = wceq 1372   e. wcel 2175   _Vcvv 2771   u. cun 3163   {csn 3632   {cpr 3633   U.cuni 3849

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850

This theorem is referenced by:  unisng  3866  uniintsnr  3920  unisuc  4459  op1sta  5163  op2nda  5166  elxp4  5169  uniabio  5241  iotass  5248  en1bg  6891  zrhval2  14352