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Theorem rex0 3350
Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rex0 ¬ ∃𝑥 ∈ ∅ 𝜑

Proof of Theorem rex0
StepHypRef Expression
1 noel 3337 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 620 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2501 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1465  wrex 2394  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by:  0iun  3840  finexdc  6764  0ct  6960  exfzdc  9985
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