Theorem 0nelelxp

Description: A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008)

	Ref	Expression
Assertion	0nelelxp	⊢ ( 𝐶 ∈ ( 𝐴 × 𝐵 ) → ¬ ∅ ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elxp	⊢ ( 𝐶 ∈ ( 𝐴 × 𝐵 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐶 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) )
2		0nelop	⊢ ¬ ∅ ∈ ⟨ 𝑥 , 𝑦 ⟩
3		eleq2	⊢ ( 𝐶 = ⟨ 𝑥 , 𝑦 ⟩ → ( ∅ ∈ 𝐶 ↔ ∅ ∈ ⟨ 𝑥 , 𝑦 ⟩ ) )
4	2 3	mtbiri	⊢ ( 𝐶 = ⟨ 𝑥 , 𝑦 ⟩ → ¬ ∅ ∈ 𝐶 )
5	4	adantr	⊢ ( ( 𝐶 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ¬ ∅ ∈ 𝐶 )
6	5	exlimivv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝐶 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ¬ ∅ ∈ 𝐶 )
7	1 6	sylbi	⊢ ( 𝐶 ∈ ( 𝐴 × 𝐵 ) → ¬ ∅ ∈ 𝐶 )