Theorem reu0 4314

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 4313	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3371	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 199	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3  ∃wrex 3086  ∃!wreu 3365  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-dif 3907  df-nul 4286

This theorem is referenced by:  join0  18435  meet0  18436