Theorem unisng 3618

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Assertion

Ref	Expression
unisng	|- ( A e. V -> U. { A } = A )

Proof of Theorem unisng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3409	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3612	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2095	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2604	. . 3 |- x e. _V
6	5	unisn 3617	. 2 |- U. { x } = x
7	4, 6	vtoclg 2658	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {csn 3398   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602

This theorem is referenced by:  dfnfc2  3619  unisucg  4169  unisn3  4198  opswapg  4827  funfvdm  5257