Theorem reu0 4318

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 4317	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3431	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 199	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3  ∃wrex 3139  ∃!wreu 3140  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-dif 3939  df-nul 4292

This theorem is referenced by:  meet0  17747  join0  17748