 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-6 Structured version   Visualization version   GIF version

Axiom ax-6 1886
 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 axc10 2250 and ax6fromc10 34000. A more convenient form of this axiom is ax6e 2248, 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-6 1886 can be proved from the weaker version ax6v 1887 requiring that the variables be distinct; see theorem ax6 2249. ax-6 1886 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 4776. Except by ax6v 1887, this axiom should not be referenced directly. Instead, use theorem ax6 2249. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
Assertion
Ref Expression
ax-6 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Detailed syntax breakdown of Axiom ax-6
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
2 vy . . . . 5 setvar 𝑦
31, 2weq 1872 . . . 4 wff 𝑥 = 𝑦
43wn 3 . . 3 wff ¬ 𝑥 = 𝑦
54, 1wal 1479 . 2 wff 𝑥 ¬ 𝑥 = 𝑦
65wn 3 1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff setvar class This axiom is referenced by:  ax6v  1887
 Copyright terms: Public domain W3C validator