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Theorem ndisj 4328
Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
ndisj ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ndisj
StepHypRef Expression
1 n0 4307 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 elin 3927 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32exbii 1851 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
41, 3bitri 275 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  wne 2940  cin 3910  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-dif 3914  df-in 3918  df-nul 4284
This theorem is referenced by:  xpcogend  14865  metsscmetcld  24695  0pssin  42131
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