Theorem unisn 3788

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 3574	. . 3 |- { A } = { A , A }
2	1	unieqi 3782	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3786	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3250	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2182	1 |- U. { A } = A

Syntax hints:   = wceq 1335   e. wcel 2128   _Vcvv 2712   u. cun 3100   {csn 3560   {cpr 3561   U.cuni 3772

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-uni 3773

This theorem is referenced by:  unisng  3789  uniintsnr  3843  unisuc  4373  op1sta  5067  op2nda  5070  elxp4  5073  uniabio  5145  iotass  5152  en1bg  6745