Theorem uniintsnr 3815

Description: The union and intersection of a singleton are equal. See also eusn 3605. (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
uniintsnr	|- ( E. x A = { x } -> U. A = |^| A )

Distinct variable group:   x, A

Proof of Theorem uniintsnr

Step	Hyp	Ref	Expression
1		vex 2692	. . . 4 |- x e. _V
2	1	unisn 3760	. . 3 |- U. { x } = x
3		unieq 3753	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3782	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3814	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2189	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2199	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1578	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1332   E.wex 1469   {csn 3532   U.cuni 3744   |^|cint 3779

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780

This theorem is referenced by:  uniintabim  3816