Theorem pssdifn0

Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994)

	Ref	Expression
Assertion	pssdifn0	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ∖ 𝐴 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		ssdif0	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐵 ∖ 𝐴 ) = ∅ )
2		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
3	2	simplbi2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐴 → 𝐴 = 𝐵 ) )
4	1 3	syl5bir	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐵 ∖ 𝐴 ) = ∅ → 𝐴 = 𝐵 ) )
5	4	necon3d	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ≠ 𝐵 → ( 𝐵 ∖ 𝐴 ) ≠ ∅ ) )
6	5	imp	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ∖ 𝐴 ) ≠ ∅ )