Theorem unisng 3866

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 3643	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3860	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2219	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2774	. . 3 |- x e. _V
6	5	unisn 3865	. 2 |- U. { x } = x
7	4, 6	vtoclg 2832	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   {csn 3632   U.cuni 3849

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850

This theorem is referenced by:  dfnfc2  3867  unisucg  4460  unisn3  4491  opswapg  5168  funfvdm  5641  en2other2  7303  lspuni0  14157  lss0v  14163