Theorem dtruex 4683

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4306 can also be summarized as "at least two sets exist", the difference is that dtruarb 4306 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 2818	. . . . 5 |- y e. _V
2	1	snex 4300	. . . 4 |- { y } e. _V
3	2	isseti 2824	. . 3 |- E. x x = { y }
4		elirrv 4672	. . . . . . 7 |- -. y e. y
5		vsnid 3723	. . . . . . . 8 |- y e. { y }
6		eleq2 2298	. . . . . . . 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 2252	. . . . . 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 677	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 698	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1651	. 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 1654	. 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 1398   E.wex 1541   e. wcel 2205   {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3215  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697

This theorem is referenced by:  dtru  4684  eunex  4685  brprcneu  5665