Theorem dtruex 4365

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4017 can also be summarized as "at least two sets exist", the difference is that dtruarb 4017 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 2622	. . . . 5 |- y e. _V
2	1	snex 4011	. . . 4 |- { y } e. _V
3	2	isseti 2627	. . 3 |- E. x x = { y }
4		elirrv 4354	. . . . . . 7 |- -. y e. y
5		vsnid 3471	. . . . . . . 8 |- y e. { y }
6		eleq2 2151	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 166	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 623	. . . . . 6 |- -. y = { y }
9		eqtr 2105	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 623	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 262	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 630	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 660	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1538	. 2 |- E. x -. y = x
15		equcom 1639	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 629	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1541	. 2 |- ( E. x -. y = x <-> E. x -. x = y )
18	14, 17	mpbi 143	1 |- E. x -. x = y

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1289   E.wex 1426   e. wcel 1438   {csn 3441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-setind 4343

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447

This theorem is referenced by:  dtru  4366  eunex  4367  brprcneu  5282