Theorem unisn 3747

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 3536	. . 3 |- { A } = { A , A }
2	1	unieqi 3741	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3745	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3214	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2162	1 |- U. { A } = A

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2681   u. cun 3064   {csn 3522   {cpr 3523   U.cuni 3731

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732

This theorem is referenced by:  unisng  3748  uniintsnr  3802  unisuc  4330  op1sta  5015  op2nda  5018  elxp4  5021  uniabio  5093  iotass  5100  en1bg  6687