Description: The Hartogs number of a set is never zero.MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | harn0 | ⊢ ( 𝑆 ∈ 𝑉 → ( har ‘ 𝑆 ) ≠ ∅ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon | ⊢ ∅ ∈ On | |
2 | 1 | a1i | ⊢ ( 𝑆 ∈ 𝑉 → ∅ ∈ On ) |
3 | 0domg | ⊢ ( 𝑆 ∈ 𝑉 → ∅ ≼ 𝑆 ) | |
4 | elharval | ⊢ ( ∅ ∈ ( har ‘ 𝑆 ) ↔ ( ∅ ∈ On ∧ ∅ ≼ 𝑆 ) ) | |
5 | 2 3 4 | sylanbrc | ⊢ ( 𝑆 ∈ 𝑉 → ∅ ∈ ( har ‘ 𝑆 ) ) |
6 | 5 | ne0d | ⊢ ( 𝑆 ∈ 𝑉 → ( har ‘ 𝑆 ) ≠ ∅ ) |