Theorem dtruex 4607

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4235 can also be summarized as "at least two sets exist", the difference is that dtruarb 4235 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 2775	. . . . 5 |- y e. _V
2	1	snex 4229	. . . 4 |- { y } e. _V
3	2	isseti 2780	. . 3 |- E. x x = { y }
4		elirrv 4596	. . . . . . 7 |- -. y e. y
5		vsnid 3665	. . . . . . . 8 |- y e. { y }
6		eleq2 2269	. . . . . . . 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 2223	. . . . . 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 1625	. 2 |- E. x -. y = x
15		equcom 1729	. . . 4 |- ( y = x <-> x = y )
16	15	notbii 670	. . 3 |- ( -. y = x <-> -. x = y )
17	16	exbii 1628	. 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 1515   e. wcel 2176   {csn 3633

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639

This theorem is referenced by:  dtru  4608  eunex  4609  brprcneu  5569