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Theorem inn0f 38268
 Description: A non-empty 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 inn0f.1 . . . 4 𝑥𝐴
2 inn0f.2 . . . 4 𝑥𝐵
31, 2nfin 3782 . . 3 𝑥(𝐴𝐵)
43n0f 3886 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
5 elin 3758 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65exbii 1764 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
7 df-rex 2902 . . 3 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
87bicomi 213 . 2 (∃𝑥(𝑥𝐴𝑥𝐵) ↔ ∃𝑥𝐴 𝑥𝐵)
94, 6, 83bitri 285 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738   ≠ wne 2780  ∃wrex 2897   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by:  inn0  38270
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