Theorem ordssun

Description: Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003)

	Ref	Expression
Assertion	ordssun	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ordtri2or2	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) )
2		ssequn1	⊢ ( 𝐵 ⊆ 𝐶 ↔ ( 𝐵 ∪ 𝐶 ) = 𝐶 )
3		sseq2	⊢ ( ( 𝐵 ∪ 𝐶 ) = 𝐶 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 ⊆ 𝐶 ) )
4	2 3	sylbi	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 ⊆ 𝐶 ) )
5		olc	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) )
6	4 5	biimtrdi	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )
7		ssequn2	⊢ ( 𝐶 ⊆ 𝐵 ↔ ( 𝐵 ∪ 𝐶 ) = 𝐵 )
8		sseq2	⊢ ( ( 𝐵 ∪ 𝐶 ) = 𝐵 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 ⊆ 𝐵 ) )
9	7 8	sylbi	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 ⊆ 𝐵 ) )
10		orc	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) )
11	9 10	biimtrdi	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )
12	6 11	jaoi	⊢ ( ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )
13	1 12	syl	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )
14		ssun	⊢ ( ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) → 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) )
15	13 14	impbid1	⊢ ( ( Ord 𝐵 ∧ Ord 𝐶 ) → ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶 ) ) )