Theorem ordsseleq

Description: For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	ordsseleq	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ordelpss	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵 ) )
2	1	orbi1d	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3		sspss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) )
4	2 3	syl6rbbr	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )