Theorem nssne2

Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015)

	Ref	Expression
Assertion	nssne2	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶 ) → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶 ) )
2	1	biimpcd	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐶 ) )
3	2	necon3bd	⊢ ( 𝐴 ⊆ 𝐶 → ( ¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵 ) )
4	3	imp	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶 ) → 𝐴 ≠ 𝐵 )