Theorem unisng 3910

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 3680	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3904	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2246	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2805	. . 3 |- x e. _V
6	5	unisn 3909	. 2 |- U. { x } = x
7	4, 6	vtoclg 2864	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {csn 3669   U.cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894

This theorem is referenced by:  dfnfc2  3911  unisucg  4511  unisn3  4542  opswapg  5223  funfvdm  5709  en2other2  7406  lspuni0  14437  lss0v  14443