MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inn0f Structured version   Visualization version   GIF version

Theorem inn0f 4371
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 3967 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21exbii 1848 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
3 inn0f.1 . . . 4 𝑥𝐴
4 inn0f.2 . . . 4 𝑥𝐵
53, 4nfin 4224 . . 3 𝑥(𝐴𝐵)
65n0f 4349 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
7 df-rex 3071 . 2 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
82, 6, 73bitr4i 303 1 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1779  wcel 2108  wnfc 2890  wne 2940  wrex 3070  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-rex 3071  df-v 3482  df-dif 3954  df-in 3958  df-nul 4334
This theorem is referenced by:  inn0  4372
  Copyright terms: Public domain W3C validator