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Theorem rex0 3375
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 3362 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 635 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2522 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1480  wrex 2415  c0 3358
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-nul 3359
This theorem is referenced by:  0iun  3865  finexdc  6789  0ct  6985  exfzdc  10010
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