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Theorem inn0f 38713
 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 elin 3779 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21exbii 1772 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
3 inn0f.1 . . . 4 𝑥𝐴
4 inn0f.2 . . . 4 𝑥𝐵
53, 4nfin 3803 . . 3 𝑥(𝐴𝐵)
65n0f 3908 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
7 df-rex 2918 . 2 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
82, 6, 73bitr4i 292 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1701   ∈ wcel 1992  Ⅎwnfc 2754   ≠ wne 2796  ∃wrex 2913   ∩ cin 3559  ∅c0 3896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-in 3567  df-nul 3897 This theorem is referenced by:  inn0  38715
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