Theorem reu0 4312

Description: Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)

Assertion

Ref	Expression
reu0	⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

Proof of Theorem reu0

Step	Hyp	Ref	Expression
1		rex0 4311	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3373	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 200	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

This proof depends on syntax axioms:  ¬ wn 3  ∃wrex 3090  ∃!wreu 3367  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-dif 4050  df-nul 4282

This theorem is used by:  join0  18627  meet0  18628