Axiom ax-9 1102

Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom in effect tells us that at least one thing exists. In this form (not requiring that x and y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9 1110 and ax9a 1111. A more convenient form of this axiom is a9e 1112, which has additional remarks.

Raph Levien proved the independence of this axiom from the others on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

Assertion

Ref	Expression
ax-9	|- -. A.x -. x = y

Detailed syntax breakdown of Axiom ax-9

Step	Hyp	Ref	Expression
1		vx	. . . . . 6 set x
2	1	cv 1098	. . . . 5 class x
3		vy	. . . . . 6 set y
4	3	cv 1098	. . . . 5 class y
5	2, 4	wceq 1099	. . . 4 wff x = y
6	5	wn 2	. . 3 wff -. x = y
7	6, 1	wal 950	. 2 wff A.x -. x = y
8	7	wn 2	1 wff -. A.x -. x = y

This axiom is referenced by:  ax9 1110

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