Theorem dtruex 4657

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4281 can also be summarized as "at least two sets exist", the difference is that dtruarb 4281 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 2805	. . . . 5 |- y e. _V
2	1	snex 4275	. . . 4 |- { y } e. _V
3	2	isseti 2811	. . 3 |- E. x x = { y }
4		elirrv 4646	. . . . . . 7 |- -. y e. y
5		vsnid 3701	. . . . . . . 8 |- y e. { y }
6		eleq2 2295	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 168	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 668	. . . . . 6 |- -. y = { y }
9		eqtr 2249	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 668	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 266	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 676	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 697	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1650	. 2 |- E. x -. y = x
15		equcom 1754	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 674	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1653	. 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 1397   E.wex 1540   e. wcel 2202   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675

This theorem is referenced by:  dtru  4658  eunex  4659  brprcneu  5632