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Axiom ax-bndl 1502
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 1500 as can be seen at axi12 1507. Whether ax-bndl 1502 can be proved from the remaining axioms including ax-i12 1500 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 1496 . . 3 wff 𝑧 = 𝑥
43, 1wal 1346 . 2 wff 𝑧 𝑧 = 𝑥
5 vy . . . . 5 setvar 𝑦
61, 5weq 1496 . . . 4 wff 𝑧 = 𝑦
76, 1wal 1346 . . 3 wff 𝑧 𝑧 = 𝑦
82, 5weq 1496 . . . . . 6 wff 𝑥 = 𝑦
98, 1wal 1346 . . . . . 6 wff 𝑧 𝑥 = 𝑦
108, 9wi 4 . . . . 5 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1110, 1wal 1346 . . . 4 wff 𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1211, 2wal 1346 . . 3 wff 𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
137, 12wo 703 . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
144, 13wo 703 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  1507  nfsbxy  1935  nfsbxyt  1936  sbcomxyyz  1965  dvelimor  2011  oprabidlem  5884
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