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Theorem rex0 3265
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 3255 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 585 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2428 1 ¬ ∃𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1409  wrex 2324  c0 3251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-nul 3252
This theorem is referenced by:  0iun  3741
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