MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-9 Structured version   GIF version

Axiom ax-9 1669
Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is 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 ax9o 1958 and ax9from9o 2232. A more convenient form of this axiom is a9e 1956, which has additional remarks.

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

ax-9 1669 can be proved from the weaker version ax9v 1670 requiring that the variables be distinct; see theorem ax9 1957.

ax-9 1669 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4365.

Except by ax9v 1670, this axiom should not be referenced directly. Instead, use theorem ax9 1957. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Detailed syntax breakdown of Axiom ax-9
StepHypRef Expression
1 vx . . . . 5 set 𝑥
2 vy . . . . 5 set 𝑦
31, 2weq 1655 . . . 4 wff 𝑥 = 𝑦
43wn 3 . . 3 wff ¬ 𝑥 = 𝑦
54, 1wal 1550 . 2 wff 𝑥 ¬ 𝑥 = 𝑦
65wn 3 1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
This axiom is referenced by:  ax9v  1670
  Copyright terms: Public domain W3C validator