Theorem dtruex 4482

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4123 can also be summarized as "at least two sets exist", the difference is that dtruarb 4123 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 2692	. . . . 5 |- y e. _V
2	1	snex 4117	. . . 4 |- { y } e. _V
3	2	isseti 2697	. . 3 |- E. x x = { y }
4		elirrv 4471	. . . . . . 7 |- -. y e. y
5		vsnid 3564	. . . . . . . 8 |- y e. { y }
6		eleq2 2204	. . . . . . . 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 2158	. . . . . 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 1683	. . . 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 1332   E.wex 1469   e. wcel 1481   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538

This theorem is referenced by:  dtru  4483  eunex  4484  brprcneu  5422