Theorem reu0 4313

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 4312	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3354	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 197	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3  ∃wrex 3060  ∃!wreu 3348  ∅c0 4285

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-dif 3904  df-nul 4286

This theorem is referenced by:  join0  18326  meet0  18327