Theorem inn0 4328

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 2925	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2925	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	inn0f 4327	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∩ cin 3905  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-rex 3090  df-v 3457  df-dif 3909  df-in 3913  df-nul 4288

This theorem is referenced by:  ufdprmidl  33809  sswfaxreg  45676  qinioo  46231