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Theorem wl-nax6im 34922
Description: The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 1964 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 892, for example. Whatever it is, we start out with the contraposition of ax-6 1970, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain of discourse could be. (Contributed by Wolf Lammen, 12-Mar-2023.)
Hypothesis
Ref Expression
wl-nax6im.1 (¬ ∃𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
wl-nax6im (¬ ∃𝑥⊤ → 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-nax6im
StepHypRef Expression
1 trud 1548 . . 3 (𝑥 = 𝑦 → ⊤)
21eximi 1836 . 2 (∃𝑥 𝑥 = 𝑦 → ∃𝑥⊤)
3 wl-nax6im.1 . 2 (¬ ∃𝑥 𝑥 = 𝑦𝜑)
42, 3nsyl5 162 1 (¬ ∃𝑥⊤ → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wtru 1539  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-tru 1541  df-ex 1782
This theorem is referenced by: (None)
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