Metamath Proof Explorer


Theorem harn0

Description: The Hartogs number of a set is never zero.MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015)

Ref Expression
Assertion harn0 ( 𝑆𝑉 → ( har ‘ 𝑆 ) ≠ ∅ )

Proof

Step Hyp Ref Expression
1 0elon ∅ ∈ On
2 1 a1i ( 𝑆𝑉 → ∅ ∈ On )
3 0domg ( 𝑆𝑉 → ∅ ≼ 𝑆 )
4 elharval ( ∅ ∈ ( har ‘ 𝑆 ) ↔ ( ∅ ∈ On ∧ ∅ ≼ 𝑆 ) )
5 2 3 4 sylanbrc ( 𝑆𝑉 → ∅ ∈ ( har ‘ 𝑆 ) )
6 5 ne0d ( 𝑆𝑉 → ( har ‘ 𝑆 ) ≠ ∅ )