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Theorem reu0 4315
Description: Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
Assertion
Ref Expression
reu0 ¬ ∃!𝑥 ∈ ∅ 𝜑

Proof of Theorem reu0
StepHypRef Expression
1 rex0 4314 . 2 ¬ ∃𝑥 ∈ ∅ 𝜑
2 reurex 3429 . 2 (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑)
31, 2mto 198 1 ¬ ∃!𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wrex 3136  ∃!wreu 3137  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-dif 3936  df-nul 4289
This theorem is referenced by:  meet0  17735  join0  17736
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