Theorem unisn 3951

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 3723	. . 3 |- { A } = { A , A }
2	1	unieqi 3945	. 2 |- U. { A } = U. { A , A }
3		unisn.1	. . 3 |- A e. _V
4	3, 3	unipr 3949	. 2 |- U. { A , A } = ( A u. A )
5		unidm 3372	. 2 |- ( A u. A ) = A
6	2, 4, 5	3eqtri 2263	1 |- U. { A } = A

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   {csn 3709   {cpr 3710   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-uni 3936

This theorem is used by:  unisng  3952  uniintsnr  4006  unisuc  4558  op1sta  5269  op2nda  5272  elxp4  5275  uniabio  5348  iotass  5355  en1bg  7087  zrhval2  14954