Theorem 0pssin 44525

Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)

Assertion

Ref	Expression
0pssin	⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem 0pssin

Step	Hyp	Ref	Expression
1		0pss 4366	. 2 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ≠ ∅)
2		ndisj 4324	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3	1, 2	bitri 278	1 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957   ∩ cin 3903   ⊊ wpss 3905  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286

This theorem is used by: (None)