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Theorem rex0 4271
 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 4247 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 119 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 3228 1 ¬ ∃𝑥 ∈ ∅ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2111  ∃wrex 3107  ∅c0 4243 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-dif 3884  df-nul 4244 This theorem is referenced by:  reu0  4272  rmo0  4273  0iun  4950  sup0riota  8920  cfeq0  9674  cfsuc  9675  hashge2el2difr  13842  cshws0  16434  dya2iocuni  31687  eulerpartlemgh  31782  0qs  35822  pmapglb2xN  37108  elpadd0  37145  sprsymrelfvlem  44068
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