Theorem unisng 3753

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 3538	. . . 4 |- ( x = A -> { x } = { A } )
2	1	unieqd 3747	. . 3 |- ( x = A -> U. { x } = U. { A } )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2154	. 2 |- ( x = A -> ( U. { x } = x <-> U. { A } = A ) )
5		vex 2689	. . 3 |- x e. _V
6	5	unisn 3752	. 2 |- U. { x } = x
7	4, 6	vtoclg 2746	1 |- ( A e. V -> U. { A } = A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   {csn 3527   U.cuni 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737

This theorem is referenced by:  dfnfc2  3754  unisucg  4336  unisn3  4366  opswapg  5025  funfvdm  5484  en2other2  7052