Theorem uniintsnr 3978

Description: The union and intersection of a singleton are equal. See also eusn 3758. (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 2815	. . . 4 |- x e. _V
2	1	unisn 3923	. . 3 |- U. { x } = x
3		unieq 3916	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3945	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3977	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2281	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2291	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1647	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   {csn 3682   U.cuni 3907   |^|cint 3942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-sn 3688  df-pr 3689  df-uni 3908  df-int 3943

This theorem is referenced by:  uniintabim  3979