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Theorem ndisj 4375
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 4358 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 elin 3978 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32exbii 1844 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
41, 3bitri 275 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1775  wcel 2105  wne 2937  cin 3961  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965  df-in 3969  df-nul 4339
This theorem is referenced by:  xpcogend  15009  metsscmetcld  25362  0pssin  43760
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