Theorem unisn 3866

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 3647	. . 3 |- { A } = { A , A }
2	1	unieqi 3860	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3864	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3316	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2230	1 |- U. { A } = A

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   u. cun 3164   {csn 3633   {cpr 3634   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851

This theorem is referenced by:  unisng  3867  uniintsnr  3921  unisuc  4460  op1sta  5164  op2nda  5167  elxp4  5170  uniabio  5242  iotass  5249  en1bg  6892  zrhval2  14381