Theorem inn0 4337

Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Assertion

Ref	Expression
inn0	⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem inn0

Step	Hyp	Ref	Expression
1		nfcv 2892	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2892	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	inn0f 4336	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   ∩ cin 3915  ∅c0 4298

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-rex 3055  df-v 3452  df-dif 3919  df-in 3923  df-nul 4299

This theorem is referenced by:  ufdprmidl  33518  sswfaxreg  44970  qinioo  45526