Theorem uniintsnr 3746

Description: The union and intersection of a singleton are equal. See also eusn 3536. (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 2636	. . . 4 |- x e. _V
2	1	unisn 3691	. . 3 |- U. { x } = x
3		unieq 3684	. . 3 |- ( A = { x } -> U. A = U. { x } )
4		inteq 3713	. . . 4 |- ( A = { x } -> |^| A = |^| { x } )
5	1	intsn 3745	. . . 4 |- |^| { x } = x
6	4, 5	syl6eq 2143	. . 3 |- ( A = { x } -> |^| A = x )
7	2, 3, 6	3eqtr4a 2153	. 2 |- ( A = { x } -> U. A = |^| A )
8	7	exlimiv 1541	1 |- ( E. x A = { x } -> U. A = |^| A )

Syntax hints:   -> wi 4   = wceq 1296   E.wex 1433   {csn 3466   U.cuni 3675   |^|cint 3710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711

This theorem is referenced by:  uniintabim  3747