Theorem dtruex 4625

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4251 can also be summarized as "at least two sets exist", the difference is that dtruarb 4251 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 2779	. . . . 5 |- y e. _V
2	1	snex 4245	. . . 4 |- { y } e. _V
3	2	isseti 2785	. . 3 |- E. x x = { y }
4		elirrv 4614	. . . . . . 7 |- -. y e. y
5		vsnid 3675	. . . . . . . 8 |- y e. { y }
6		eleq2 2271	. . . . . . . 8 |- ( y = { y } -> ( y e. y <-> y e. { y } ) )
7	5, 6	mpbiri 168	. . . . . . 7 |- ( y = { y } -> y e. y )
8	4, 7	mto 664	. . . . . 6 |- -. y = { y }
9		eqtr 2225	. . . . . 6 |- ( ( y = x /\ x = { y } ) -> y = { y } )
10	8, 9	mto 664	. . . . 5 |- -. ( y = x /\ x = { y } )
11		ancom 266	. . . . 5 |- ( ( y = x /\ x = { y } ) <-> ( x = { y } /\ y = x ) )
12	10, 11	mtbi 672	. . . 4 |- -. ( x = { y } /\ y = x )
13	12	imnani 693	. . 3 |- ( x = { y } -> -. y = x )
14	3, 13	eximii 1626	. 2 |- E. x -. y = x
15		equcom 1730	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 670	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1629	. 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 1373   E.wex 1516   e. wcel 2178   {csn 3643

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649

This theorem is referenced by:  dtru  4626  eunex  4627  brprcneu  5592