Theorem inn0 4352

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 2897	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	inn0f 4351	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2107   ≠ wne 2931  ∃wrex 3059   ∩ cin 3930  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-rex 3060  df-v 3465  df-dif 3934  df-in 3938  df-nul 4314

This theorem is referenced by:  ufdprmidl  33504  sswfaxreg  44961  qinioo  45505