Description: Any subset of the empty set is empty. Theorem 5 of Suppes p. 23 and its converse. (Contributed by NM, 17-Sep-2003)
Ref | Expression | ||
---|---|---|---|
Assertion | ss0b | ⊢ ( 𝐴 ⊆ ∅ ↔ 𝐴 = ∅ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss | ⊢ ∅ ⊆ 𝐴 | |
2 | eqss | ⊢ ( 𝐴 = ∅ ↔ ( 𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴 ) ) | |
3 | 1 2 | mpbiran2 | ⊢ ( 𝐴 = ∅ ↔ 𝐴 ⊆ ∅ ) |
4 | 3 | bicomi | ⊢ ( 𝐴 ⊆ ∅ ↔ 𝐴 = ∅ ) |