This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ontr2

Description: Transitive law for ordinal numbers. Exercise 3 of TakeutiZaring p. 40. (Contributed by NM, 6-Nov-2003)

Ref Expression
Assertion ontr2 ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 eloni ⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2 eloni ⊢ ( 𝐶 ∈ On → Ord 𝐶 )
3 ordtr2 ⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )