Theorem unisng 3947

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 3716	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3941	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2253	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2824	. . 3 |- x e. _V
6	5	unisn 3946	. 2 |- U. { x } = x
7	4, 6	vtoclg 2883	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {csn 3705   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931

This theorem is referenced by:  dfnfc2  3948  unisucg  4554  unisn3  4586  opswapg  5269  funfvdm  5760  en2other2  7538  lspuni0  14733  lss0v  14739