Theorem unisng 3806

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 3587	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3800	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2180	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2729	. . 3 |- x e. _V
6	5	unisn 3805	. 2 |- U. { x } = x
7	4, 6	vtoclg 2786	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   {csn 3576   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by:  dfnfc2  3807  unisucg  4392  unisn3  4423  opswapg  5090  funfvdm  5549  en2other2  7152