Theorem dtruex 4701

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4323 can also be summarized as "at least two sets exist", the difference is that dtruarb 4323 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 2824	. . . . 5 |- y e. _V
2	1	snex 4317	. . . 4 |- { y } e. _V
3	2	isseti 2830	. . 3 |- E. x x = { y }
4		elirrv 4690	. . . . . . 7 |- -. y e. y
5		vsnid 3737	. . . . . . . 8 |- y e. { y }
6		eleq2 2302	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 168	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 672	. . . . . 6 |- -. y = { y }
9		eqtr 2256	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 672	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 266	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 681	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 702	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1655	. 2 |- E. x -. y = x
15		equcom 1758	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 678	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1658	. 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 1402   E.wex 1545   e. wcel 2209   {csn 3705

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711

This theorem is referenced by:  dtru  4702  eunex  4703  brprcneu  5683