Theorem ordsssuc

Description: An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003)

	Ref	Expression
Assertion	ordsssuc	⊢ ( ( 𝐴 ∈ On ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		ordsseleq	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ On ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4		elsucg	⊢ ( 𝐴 ∈ On → ( 𝐴 ∈ suc 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ On ∧ Ord 𝐵 ) → ( 𝐴 ∈ suc 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
6	3 5	bitr4d	⊢ ( ( 𝐴 ∈ On ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵 ) )