Theorem inn0 44275

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 2895	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2895	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	inn0f 44273	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 205   ∈ wcel 2098   ≠ wne 2932  ∃wrex 3062   ∩ cin 3940  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-nul 4316

This theorem is referenced by:  qinioo  44758