Theorem uniintsnr 3867

Description: The union and intersection of a singleton are equal. See also eusn 3657. (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 2733	. . . 4 |- x e. _V
2	1	unisn 3812	. . 3 |- U. { x } = x
3		unieq 3805	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3834	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3866	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2219	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2229	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1591	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1348   E.wex 1485   {csn 3583   U.cuni 3796   |^|cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832

This theorem is referenced by:  uniintabim  3868