Theorem uniintsnr 3910

Description: The union and intersection of a singleton are equal. See also eusn 3696. (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 2766	. . . 4 |- x e. _V
2	1	unisn 3855	. . 3 |- U. { x } = x
3		unieq 3848	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3877	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3909	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2245	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2255	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1612	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1506   {csn 3622   U.cuni 3839   |^|cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875

This theorem is referenced by:  uniintabim  3911