Theorem uniintsnr 3807

Description: The union and intersection of a singleton are equal. See also eusn 3597. (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 2689	. . . 4 |- x e. _V
2	1	unisn 3752	. . 3 |- U. { x } = x
3		unieq 3745	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3774	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3806	. . . 4 |- |^| { x } = x
6	4, 5	syl6eq 2188	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2198	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1577	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1331   E.wex 1468   {csn 3527   U.cuni 3736   |^|cint 3771

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-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772

This theorem is referenced by:  uniintabim  3808