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Theorem ndisj 4266
 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 4245 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 elin 3874 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32exbii 1849 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
41, 3bitri 278 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951   ∩ cin 3857  ∅c0 4225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-dif 3861  df-in 3865  df-nul 4226 This theorem is referenced by:  xpcogend  14381  metsscmetcld  24015  0pssin  40845
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