Theorem uniintsnr 3672

Description: The union and intersection of a singleton are equal. See also eusn 3466. (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 2604	. . . 4 |- x e. _V
2	1	unisn 3617	. . 3 |- U. { x } = x
3		unieq 3610	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3639	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3671	. . . 4 |- |^| { x } = x
6	4, 5	syl6eq 2129	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2139	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1529	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   {csn 3398   U.cuni 3601   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637

This theorem is referenced by:  uniintabim  3673