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Theorem rex0 4314
Description: Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rex0 ¬ ∃𝑥 ∈ ∅ 𝜑

Proof of Theorem rex0
StepHypRef Expression
1 noel 4293 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 119 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 3266 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2105  wrex 3136  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-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-dif 3936  df-nul 4289
This theorem is referenced by:  reu0  4315  rmo0  4316  0iun  4977  sup0riota  8917  cfeq0  9666  cfsuc  9667  hashge2el2difr  13827  cshws0  16423  dya2iocuni  31440  eulerpartlemgh  31535  0qs  35502  pmapglb2xN  36788  elpadd0  36825  sprsymrelfvlem  43529
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