Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of BellMachover p. 469. (Contributed by NM, 26-Oct-2003)
Ref | Expression | ||
---|---|---|---|
Assertion | onelon | ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni | ⊢ ( 𝐴 ∈ On → Ord 𝐴 ) | |
2 | ordelon | ⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On ) | |
3 | 1 2 | sylan | ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On ) |