Theorem uniintsnr 3860

Description: The union and intersection of a singleton are equal. See also eusn 3650. (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 2729	. . . 4 |- x e. _V
2	1	unisn 3805	. . 3 |- U. { x } = x
3		unieq 3798	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3827	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3859	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2215	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2225	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1586	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1343   E.wex 1480   {csn 3576   U.cuni 3789   |^|cint 3824

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825

This theorem is referenced by:  uniintabim  3861