Theorem dtruarb 3962

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4302 in which we are given a set y and go from there to a set x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)

Assertion

Ref	Expression
dtruarb	|- E. x E. y -. x = y

Distinct variable group:   x, y

Proof of Theorem dtruarb

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 3952	. . 3 |- E. x z e. x
2		ax-nul 3904	. . . 4 |- E. y A. z -. z e. y
3		sp 1441	. . . 4 |- ( A. z -. z e. y -> -. z e. y )
4	2, 3	eximii 1533	. . 3 |- E. y -. z e. y
5		eeanv 1848	. . 3 |- ( E. x E. y ( z e. x /\ -. z e. y ) <-> ( E. x z e. x /\ E. y -. z e. y ) )
6	1, 4, 5	mpbir2an 883	. 2 |- E. x E. y ( z e. x /\ -. z e. y )
7		nelneq2 2180	. . 3 |- ( ( z e. x /\ -. z e. y ) -> -. x = y )
8	7	2eximi 1532	. 2 |- ( E. x E. y ( z e. x /\ -. z e. y ) -> E. x E. y -. x = y )
9	6, 8	ax-mp 7	1 |- E. x E. y -. x = y

Syntax hints:   -. wn 3   /\ wa 102   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077

This theorem is referenced by: (None)