Theorem uniintsnr 3906

Description: The union and intersection of a singleton are equal. See also eusn 3692. (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 2763	. . . 4 |- x e. _V
2	1	unisn 3851	. . 3 |- U. { x } = x
3		unieq 3844	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3873	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3905	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2242	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2252	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1609	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1503   {csn 3618   U.cuni 3835   |^|cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871

This theorem is referenced by:  uniintabim  3907