Theorem dtruarb 4170

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 4536 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 4157	. . 3 |- E. x z e. x
2		ax-nul 4108	. . . 4 |- E. y A. z -. z e. y
3		sp 1499	. . . 4 |- ( A. z -. z e. y -> -. z e. y )
4	2, 3	eximii 1590	. . 3 |- E. y -. z e. y
5		eeanv 1920	. . 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 932	. 2 |- E. x E. y ( z e. x /\ -. z e. y )
7		nelneq2 2268	. . 3 |- ( ( z e. x /\ -. z e. y ) -> -. x = y )
8	7	2eximi 1589	. 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 103   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-13 2138  ax-14 2139  ax-ext 2147  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161

This theorem is referenced by: (None)