Theorem uniintsnr 3921

Description: The union and intersection of a singleton are equal. See also eusn 3707. (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 2775	. . . 4 |- x e. _V
2	1	unisn 3866	. . 3 |- U. { x } = x
3		unieq 3859	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3888	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3920	. . . 4 |- |^| { x } = x
6	4, 5	eqtrdi 2254	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2264	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1621	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1373   E.wex 1515   {csn 3633   U.cuni 3850   |^|cint 3885

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886

This theorem is referenced by:  uniintabim  3922