Theorem unisng 3904

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 3677	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3898	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2244	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2802	. . 3 |- x e. _V
6	5	unisn 3903	. 2 |- U. { x } = x
7	4, 6	vtoclg 2861	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {csn 3666   U.cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888

This theorem is referenced by:  dfnfc2  3905  unisucg  4504  unisn3  4535  opswapg  5214  funfvdm  5696  en2other2  7370  lspuni0  14382  lss0v  14388