Theorem unisng 3931

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 3700	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3925	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2247	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2816	. . 3 |- x e. _V
6	5	unisn 3930	. 2 |- U. { x } = x
7	4, 6	vtoclg 2875	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   {csn 3689   U.cuni 3914

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-uni 3915

This theorem is referenced by:  dfnfc2  3932  unisucg  4535  unisn3  4566  opswapg  5249  funfvdm  5740  en2other2  7499  lspuni0  14572  lss0v  14578