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Theorem inn0 44430
Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Assertion
Ref Expression
inn0 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem inn0
StepHypRef Expression
1 nfcv 2899 . 2 𝑥𝐴
2 nfcv 2899 . 2 𝑥𝐵
31, 2inn0f 44428 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  wne 2936  wrex 3066  cin 3944  c0 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-nul 4320
This theorem is referenced by:  qinioo  44911
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