Theorem dtruarb 4281

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 4657 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 4268	. . 3 |- E. x z e. x
2		ax-nul 4215	. . . 4 |- E. y A. z -. z e. y
3		sp 1559	. . . 4 |- ( A. z -. z e. y -> -. z e. y )
4	2, 3	eximii 1650	. . 3 |- E. y -. z e. y
5		eeanv 1985	. . 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 950	. 2 |- E. x E. y ( z e. x /\ -. z e. y )
7		nelneq2 2333	. . 3 |- ( ( z e. x /\ -. z e. y ) -> -. x = y )
8	7	2eximi 1649	. 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 1395   E.wex 1540

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 619  ax-in2 620  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-clel 2227

This theorem is referenced by: (None)