Theorem dtruarb 4225

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 4596 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 4212	. . 3 |- E. x z e. x
2		ax-nul 4160	. . . 4 |- E. y A. z -. z e. y
3		sp 1525	. . . 4 |- ( A. z -. z e. y -> -. z e. y )
4	2, 3	eximii 1616	. . 3 |- E. y -. z e. y
5		eeanv 1951	. . 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 944	. 2 |- E. x E. y ( z e. x /\ -. z e. y )
7		nelneq2 2298	. . 3 |- ( ( z e. x /\ -. z e. y ) -> -. x = y )
8	7	2eximi 1615	. 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 1362   E.wex 1506

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192

This theorem is referenced by: (None)