Theorem 0pssin

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

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

Proof

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