Theorem unisng 3813

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 3594	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3807	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2185	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2733	. . 3 |- x e. _V
6	5	unisn 3812	. 2 |- U. { x } = x
7	4, 6	vtoclg 2790	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {csn 3583   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  dfnfc2  3814  unisucg  4399  unisn3  4430  opswapg  5097  funfvdm  5559  en2other2  7173