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Theorem inn0f 44222
Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
inn0f.1 𝑥𝐴
inn0f.2 𝑥𝐵
Assertion
Ref Expression
inn0f ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)

Proof of Theorem inn0f
StepHypRef Expression
1 elin 3964 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21exbii 1849 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
3 inn0f.1 . . . 4 𝑥𝐴
4 inn0f.2 . . . 4 𝑥𝐵
53, 4nfin 4216 . . 3 𝑥(𝐴𝐵)
65n0f 4342 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
7 df-rex 3070 . 2 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
82, 6, 73bitr4i 303 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1780  wcel 2105  wnfc 2882  wne 2939  wrex 3069  cin 3947  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-in 3955  df-nul 4323
This theorem is referenced by:  inn0  44224
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