Theorem dtruex 4576

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4209 can also be summarized as "at least two sets exist", the difference is that dtruarb 4209 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dtruex	|- E. x -. x = y

Distinct variable group:   x, y

Proof of Theorem dtruex

Step	Hyp	Ref	Expression
1		vex 2755	. . . . 5 |- y e. _V
2	1	snex 4203	. . . 4 |- { y } e. _V
3	2	isseti 2760	. . 3 |- E. x x = { y }
4		elirrv 4565	. . . . . . 7 |- -. y e. y
5		vsnid 3639	. . . . . . . 8 |- y e. { y }
6		eleq2 2253	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 168	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 663	. . . . . 6 |- -. y = { y }
9		eqtr 2207	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 663	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 266	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 671	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 692	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1613	. 2 |- E. x -. y = x
15		equcom 1717	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 669	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1616	. 2 |- ( E. x -. y = x <-> E. x -. x = y )
18	14, 17	mpbi 145	1 |- E. x -. x = y

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   {csn 3607

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-in1 615  ax-in2 616  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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613

This theorem is referenced by:  dtru  4577  eunex  4578  brprcneu  5527