ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-bndl GIF version

Axiom ax-bndl 1440
Description: Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as 𝑧𝑧 = 𝑥 ∨ ¬ ∀𝑧𝑧 = 𝑥. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object. 𝑧𝑧 = 𝑥 holds if 𝑧 and 𝑥 are the same variable, likewise for 𝑧 and 𝑦, and 𝑥𝑧(𝑥 = 𝑦 → ∀𝑧𝑥 = 𝑦) holds if 𝑧 is distinct from the others (and the universe has at least two objects).

As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability).

This axiom implies ax-i12 1439 as can be seen at axi12 1448. Whether ax-bndl 1440 can be proved from the remaining axioms including ax-i12 1439 is not known.

The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)

Assertion
Ref Expression
ax-bndl (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Detailed syntax breakdown of Axiom ax-bndl
StepHypRef Expression
1 vz . . . 4 setvar 𝑧
2 vx . . . 4 setvar 𝑥
31, 2weq 1433 . . 3 wff 𝑧 = 𝑥
43, 1wal 1283 . 2 wff 𝑧 𝑧 = 𝑥
5 vy . . . . 5 setvar 𝑦
61, 5weq 1433 . . . 4 wff 𝑧 = 𝑦
76, 1wal 1283 . . 3 wff 𝑧 𝑧 = 𝑦
82, 5weq 1433 . . . . . 6 wff 𝑥 = 𝑦
98, 1wal 1283 . . . . . 6 wff 𝑧 𝑥 = 𝑦
108, 9wi 4 . . . . 5 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1110, 1wal 1283 . . . 4 wff 𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1211, 2wal 1283 . . 3 wff 𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
137, 12wo 662 . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
144, 13wo 662 1 wff (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff set class
This axiom is referenced by:  axi12  1448  nfsbxy  1861  nfsbxyt  1862  sbcomxyyz  1889  dvelimor  1937  oprabidlem  5614
  Copyright terms: Public domain W3C validator