Theorem 0pssin 44963

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 4363	. 2 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ≠ ∅)
2		ndisj 4321	. 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 401  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959   ⊊ wpss 4016   ∩ cin 4092  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-nul 4282

This theorem is used by: (None)