Theorem elnelne1

Description: Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020)

	Ref	Expression
Assertion	elnelne1	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶 ) → 𝐵 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		df-nel	⊢ ( 𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶 )
2		nelne1	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) → 𝐵 ≠ 𝐶 )
3	1 2	sylan2b	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶 ) → 𝐵 ≠ 𝐶 )