Theorem unisng 3626

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 3417	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3620	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2096	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2605	. . 3 |- x e. _V
6	5	unisn 3625	. 2 |- U. { x } = x
7	4, 6	vtoclg 2659	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   {csn 3406   U.cuni 3609

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-uni 3610

This theorem is referenced by:  dfnfc2  3627  unisucg  4177  unisn3  4206  opswapg  4837  funfvdm  5268  en2other2  6522