Theorem dtruarb 4189

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 4556 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 4176	. . 3 |- E. x z e. x
2		ax-nul 4127	. . . 4 |- E. y A. z -. z e. y
3		sp 1511	. . . 4 |- ( A. z -. z e. y -> -. z e. y )
4	2, 3	eximii 1602	. . 3 |- E. y -. z e. y
5		eeanv 1932	. . 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 942	. 2 |- E. x E. y ( z e. x /\ -. z e. y )
7		nelneq2 2279	. . 3 |- ( ( z e. x /\ -. z e. y ) -> -. x = y )
8	7	2eximi 1601	. 2 |- ( E. x E. y ( z e. x /\ -. z e. y ) -> E. x E. y -. x = y )
9	6, 8	ax-mp 5	1 |- E. x E. y -. x = y

Syntax hints:   -. wn 3   /\ wa 104   A.wal 1351   E.wex 1492

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 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4127  ax-pow 4172

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173

This theorem is referenced by: (None)