Theorem inn0 4319

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 2922	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2922	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	inn0f 4318	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   ∩ cin 4089  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-rex 3087  df-v 3452  df-dif 4047  df-in 4091  df-nul 4279

This theorem is used by:  tgaaddcpbl2  29358  ufdprmidl  34119  sswfaxreg  46070  qinioo  46640