Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom 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 axc10 and ax6fromc10 . A more convenient form of this axiom is ax6e , 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 https://us.metamath.org/award2003.html .
ax-6 can be proved from the weaker version ax6v requiring that the variables be distinct; see Theorem ax6 .
ax-6 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 ax6vsep .
Except by ax6v , this axiom should not be referenced directly. Instead, use Theorem ax6 . (Contributed by NM, 10-Jan-1993) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-6 | ⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | ⊢ 𝑥 | |
| 1 | 0 | cv | ⊢ 𝑥 |
| 2 | vy | ⊢ 𝑦 | |
| 3 | 2 | cv | ⊢ 𝑦 |
| 4 | 1 3 | wceq | ⊢ 𝑥 = 𝑦 |
| 5 | 4 | wn | ⊢ ¬ 𝑥 = 𝑦 |
| 6 | 5 0 | wal | ⊢ ∀ 𝑥 ¬ 𝑥 = 𝑦 |
| 7 | 6 | wn | ⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 |