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Theorem inn0 39660
Description: A non-empty 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 2866 . 2 𝑥𝐴
2 nfcv 2866 . 2 𝑥𝐵
31, 2inn0f 39658 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2103  wne 2896  wrex 3015  cin 3679  c0 4023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-in 3687  df-nul 4024
This theorem is referenced by:  qinioo  40182
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