Theorem pssdif

Description: A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016)

	Ref	Expression
Assertion	pssdif	⊢ ( 𝐴 ⊊ 𝐵 → ( 𝐵 ∖ 𝐴 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-pss	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) )
2		pssdifn0	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ∖ 𝐴 ) ≠ ∅ )
3	1 2	sylbi	⊢ ( 𝐴 ⊊ 𝐵 → ( 𝐵 ∖ 𝐴 ) ≠ ∅ )