Theorem unisng 3881

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 3654	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3875	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2222	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2779	. . 3 |- x e. _V
6	5	unisn 3880	. 2 |- U. { x } = x
7	4, 6	vtoclg 2838	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   {csn 3643   U.cuni 3864

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865

This theorem is referenced by:  dfnfc2  3882  unisucg  4479  unisn3  4510  opswapg  5188  funfvdm  5665  en2other2  7335  lspuni0  14301  lss0v  14307