Theorem wl-nax6im 33841

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 2061 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 923, for example. Whatever it is, we start out with the contraposition of ax-6 2075, 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 could be. (Contributed by Wolf Lammen, 12-Mar-2023.)

Hypothesis

Ref	Expression
wl-nax6im.1	⊢ (¬ ∃𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
wl-nax6im	⊢ (¬ ∃𝑥⊤ → 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-nax6im

Step	Hyp	Ref	Expression
1		trud 1667	. . . 4 ⊢ (𝑥 = 𝑦 → ⊤)
2	1	eximi 1933	. . 3 ⊢ (∃𝑥 𝑥 = 𝑦 → ∃𝑥⊤)
3		wl-nax6im.1	. . 3 ⊢ (¬ ∃𝑥 𝑥 = 𝑦 → 𝜑)
4	2, 3	nsyl4 158	. 2 ⊢ (¬ 𝜑 → ∃𝑥⊤)
5	4	con1i 147	1 ⊢ (¬ ∃𝑥⊤ → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊤wtru 1657  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908

This theorem depends on definitions:  df-bi 199  df-tru 1660  df-ex 1879

This theorem is referenced by: (None)