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Theorem rex0 3431
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 3418 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 641 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2562 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2141  wrex 2449  c0 3414
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-nul 3415
This theorem is referenced by:  0iun  3928  finexdc  6876  0ct  7080  exfzdc  10183
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