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Theorem rex0 4321
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 4300 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 119 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 3274 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  wrex 3144  c0 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-ral 3148  df-rex 3149  df-dif 3943  df-nul 4296
This theorem is referenced by:  reu0  4322  rmo0  4323  0iun  4983  sup0riota  8923  cfeq0  9672  cfsuc  9673  hashge2el2difr  13834  cshws0  16430  dya2iocuni  31446  eulerpartlemgh  31541  0qs  35508  pmapglb2xN  36794  elpadd0  36831  sprsymrelfvlem  43503
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