Theorem reu0 4311

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 4310	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3372	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 200	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

This proof depends on syntax axioms:  ¬ wn 3  ∃wrex 3089  ∃!wreu 3366  ∅c0 4280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-dif 4049  df-nul 4281

This theorem is used by:  join0  18589  meet0  18590