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Theorem rex0 3300
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 3290 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 610 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2465 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1438  wrex 2360  c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-nul 3287
This theorem is referenced by:  0iun  3787  finexdc  6618  exfzdc  9651
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