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Theorem rex0 4316
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 4289 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 119 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 3076 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  wrex 3072  c0 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-dif 3912  df-nul 4282
This theorem is referenced by:  reu0  4317  rmo0  4318  0iun  5022  sup0riota  9360  cfeq0  10151  cfsuc  10152  hashge2el2difr  14334  cshws0  16934  dya2iocuni  32687  eulerpartlemgh  32782  addsid1  34273  0qs  36763  pmapglb2xN  38167  elpadd0  38204  sprsymrelfvlem  45577  ipolub00  46913
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