Theorem inn0f

Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	inn0f.1	⊢ Ⅎ 𝑥 𝐴
	inn0f.2	⊢ Ⅎ 𝑥 𝐵
Assertion	inn0f	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		inn0f.1	⊢ Ⅎ 𝑥 𝐴
2		inn0f.2	⊢ Ⅎ 𝑥 𝐵
3		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
4	3	exbii	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
5	1 2	nfin	⊢ Ⅎ 𝑥 ( 𝐴 ∩ 𝐵 )
6	5	n0f	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) )
7		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
8	4 6 7	3bitr4i	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )