Theorem ordtr2

Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	ordtr2	⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ordelord	⊢ ( ( Ord 𝐶 ∧ 𝐵 ∈ 𝐶 ) → Ord 𝐵 )
2	1	ex	⊢ ( Ord 𝐶 → ( 𝐵 ∈ 𝐶 → Ord 𝐵 ) )
3	2	ancld	⊢ ( Ord 𝐶 → ( 𝐵 ∈ 𝐶 → ( 𝐵 ∈ 𝐶 ∧ Ord 𝐵 ) ) )
4	3	anc2li	⊢ ( Ord 𝐶 → ( 𝐵 ∈ 𝐶 → ( Ord 𝐶 ∧ ( 𝐵 ∈ 𝐶 ∧ Ord 𝐵 ) ) ) )
5		ordelpss	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐵 ∈ 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )
6		sspsstr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )
7	6	expcom	⊢ ( 𝐵 ⊊ 𝐶 → ( 𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶 ) )
8	5 7	biimtrdi	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐵 ∈ 𝐶 → ( 𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶 ) ) )
9	8	expcom	⊢ ( Ord 𝐶 → ( Ord 𝐵 → ( 𝐵 ∈ 𝐶 → ( 𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶 ) ) ) )
10	9	com23	⊢ ( Ord 𝐶 → ( 𝐵 ∈ 𝐶 → ( Ord 𝐵 → ( 𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶 ) ) ) )
11	10	imp32	⊢ ( ( Ord 𝐶 ∧ ( 𝐵 ∈ 𝐶 ∧ Ord 𝐵 ) ) → ( 𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶 ) )
12	11	com12	⊢ ( 𝐴 ⊆ 𝐵 → ( ( Ord 𝐶 ∧ ( 𝐵 ∈ 𝐶 ∧ Ord 𝐵 ) ) → 𝐴 ⊊ 𝐶 ) )
13	4 12	syl9	⊢ ( Ord 𝐶 → ( 𝐴 ⊆ 𝐵 → ( 𝐵 ∈ 𝐶 → 𝐴 ⊊ 𝐶 ) ) )
14	13	impd	⊢ ( Ord 𝐶 → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ⊊ 𝐶 ) )
15	14	adantl	⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ⊊ 𝐶 ) )
16		ordelpss	⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( 𝐴 ∈ 𝐶 ↔ 𝐴 ⊊ 𝐶 ) )
17	15 16	sylibrd	⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )