Theorem unisn 3617

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 3412	. . 3 |- { A } = { A , A }
2	1	unieqi 3611	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3615	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3115	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2105	1 |- U. { A } = A

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601   u. cun 2971   {csn 3398   {cpr 3399   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602

This theorem is referenced by:  unisng  3618  uniintsnr  3672  unisuc  4168  op1sta  4822  op2nda  4825  elxp4  4828  uniabio  4897  iotass  4904  en1bg  6303