Theorem reu0 4357

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 4356	. 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2		reurex 3380	. 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
3	1, 2	mto 196	1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3  ∃wrex 3070  ∃!wreu 3374  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-dif 3950  df-nul 4322

This theorem is referenced by:  join0  18354  meet0  18355