Theorem unisn 3827

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 3608	. . 3 |- { A } = { A , A }
2	1	unieqi 3821	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3825	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3280	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2202	1 |- U. { A } = A

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2739   u. cun 3129   {csn 3594   {cpr 3595   U.cuni 3811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812

This theorem is referenced by:  unisng  3828  uniintsnr  3882  unisuc  4415  op1sta  5112  op2nda  5115  elxp4  5118  uniabio  5190  iotass  5197  en1bg  6802