Metamath Proof Explorer
Description: Transitive law for ordinal numbers. Theorem 7M(b) of Enderton p. 192.
Theorem 1.9(ii) of Schloeder p. 1. (Contributed by NM, 11-Aug-1994)
|
|
Ref |
Expression |
|
Assertion |
ontr1 |
⊢ ( 𝐶 ∈ On → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eloni |
⊢ ( 𝐶 ∈ On → Ord 𝐶 ) |
2 |
|
ordtr1 |
⊢ ( Ord 𝐶 → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝐶 ∈ On → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) ) |