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Theorem inn0f 41694
 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 3900 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21exbii 1849 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
3 inn0f.1 . . . 4 𝑥𝐴
4 inn0f.2 . . . 4 𝑥𝐵
53, 4nfin 4146 . . 3 𝑥(𝐴𝐵)
65n0f 4260 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
7 df-rex 3115 . 2 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
82, 6, 73bitr4i 306 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2112  Ⅎwnfc 2939   ≠ wne 2990  ∃wrex 3110   ∩ cin 3883  ∅c0 4246 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-in 3891  df-nul 4247 This theorem is referenced by:  inn0  41696
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