Theorem unisn 3596

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 3389	. . 3 |- { A } = { A , A }
2	1	unieqi 3590	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3594	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3086	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2064	1 |- U. { A } = A

Syntax hints:   = wceq 1243   e. wcel 1393   _Vcvv 2557   u. cun 2915   {csn 3375   {cpr 3376   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581

This theorem is referenced by:  unisng  3597  uniintsnr  3651  unisuc  4150  op1sta  4802  op2nda  4805  elxp4  4808  uniabio  4877  iotass  4884  en1bg  6280