Theorem 0pss

Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996)

	Ref	Expression
Assertion	0pss	⊢ ( ∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		0ss	⊢ ∅ ⊆ 𝐴
2		df-pss	⊢ ( ∅ ⊊ 𝐴 ↔ ( ∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴 ) )
3	1 2	mpbiran	⊢ ( ∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴 )
4		necom	⊢ ( ∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅ )
5	3 4	bitri	⊢ ( ∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅ )