Theorem unisn 3760

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 3546	. . 3 |- { A } = { A , A }
2	1	unieqi 3754	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3758	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3224	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2165	1 |- U. { A } = A

Syntax hints:   = wceq 1332   e. wcel 1481   _Vcvv 2689   u. cun 3074   {csn 3532   {cpr 3533   U.cuni 3744

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745

This theorem is referenced by:  unisng  3761  uniintsnr  3815  unisuc  4343  op1sta  5028  op2nda  5031  elxp4  5034  uniabio  5106  iotass  5113  en1bg  6702