Theorem difn0

Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017)

	Ref	Expression
Assertion	difn0	⊢ ( ( 𝐴 ∖ 𝐵 ) ≠ ∅ → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
2		ssdif0	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∖ 𝐵 ) = ∅ )
3	1 2	sylib	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∖ 𝐵 ) = ∅ )
4	3	necon3i	⊢ ( ( 𝐴 ∖ 𝐵 ) ≠ ∅ → 𝐴 ≠ 𝐵 )