Theorem dtruex 4520

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4154 can also be summarized as "at least two sets exist", the difference is that dtruarb 4154 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 2715	. . . . 5 |- y e. _V
2	1	snex 4148	. . . 4 |- { y } e. _V
3	2	isseti 2720	. . 3 |- E. x x = { y }
4		elirrv 4509	. . . . . . 7 |- -. y e. y
5		vsnid 3593	. . . . . . . 8 |- y e. { y }
6		eleq2 2221	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 167	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 652	. . . . . 6 |- -. y = { y }
9		eqtr 2175	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 652	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 264	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 660	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 681	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1582	. 2 |- E. x -. y = x
15		equcom 1686	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 658	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1585	. 2 |- ( E. x -. y = x <-> E. x -. x = y )
18	14, 17	mpbi 144	1 |- E. x -. x = y

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1335   E.wex 1472   e. wcel 2128   {csn 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-setind 4498

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567

This theorem is referenced by:  dtru  4521  eunex  4522  brprcneu  5463