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Theorem rex0 4138
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 4119 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 117 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 3180 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2157  wrex 3090  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-nul 4116
This theorem is referenced by:  0iun  4767  sup0riota  8613  cfeq0  9366  cfsuc  9367  hashge2el2difr  13512  cshws0  16136  meet0  17452  join0  17453  dya2iocuni  30861  eulerpartlemgh  30956  0qs  34626  pmapglb2xN  35793  elpadd0  35830  sprsymrelfvlem  42535
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